Sums of cubes and the Ratios Conjectures

Victor Y. Wang Department of Mathematics, Princeton University, Princeton, NJ, USA Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected]

Abstract.

Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_{1}^{3}+\dots+x_{6}^{3}=0$ in expanding regions, conditional on automorphy and GRH for certain Hasse–Weil $L$ -functions; for regions of diameter $X\geq 1$ , the bound takes the form $N\leq C(\varepsilon)X^{3+\varepsilon}$ ( $\varepsilon>0$ ). We attribute the $\varepsilon$ to several subtly interacting proof factors; we then remove the $\varepsilon$ assuming some standard number-theoretic hypotheses, mainly featuring the Ratios and Square-free Sieve Conjectures. In fact, our softest hypotheses imply conjectures of Hooley and Manin on $N$ , and show that almost all integers $a\not\equiv\pm 4\bmod{9}$ are sums of three cubes. Our fullest hypotheses are capable of proving power-saving asymptotics for $N$ , and producing almost all primes $p\not\equiv\pm 4\bmod{9}$ .

Key words and phrases:

Cubic form, circle method, rational points, Hasse–Weil

L

-functions, correlations

1991 Mathematics Subject Classification:

Primary 11D45; Secondary 11D25, 11G40, 11M50, 11P55

1. Introduction

Let $F_{0}=F_{0}(x,y,z)\colonequals x^{3}+y^{3}+z^{3}$ . For each $a\in\mathbb{Z}$ , the cubic surface $F_{0}=a$ has a fairly rich set of rational points [segre1943note]. On the other hand, Mordell has suggested that producing large, general integer solutions to $F_{0}=a$ for $a=3$ (or for any other fixed $a\in\mathbb{Z}$ ) could be as hard as “finding when an assigned sequence, e.g. $123456789$ , occurs in the decimal expansion of $\pi$ ” [mordell1953integer]*p. 505. The recent work [booker2021question] of Booker and Sutherland resolves Mordell’s specific question for $a=3$ , but the spirit of Mordell’s suggestion certainly remains.

Heath-Brown has conjectured that $F_{0}=a$ should have infinitely many solutions $(x,y,z)\in\mathbb{Z}^{3}$ for any fixed $a\not\equiv\pm 4\bmod{9}$ (see [heath1992density]*p. 623). To represent all $a\not\equiv\pm 4\bmod{9}$ even once, one must allow both positive and negative values of $x,y,z$ . The set $F_{0}(\mathbb{Z}_{\geq 0}^{3})$ has upper density $\leq\Gamma(4/3)^{3}/6=0.1186788\ldots$ in $\mathbb{Z}_{\geq 0}$ [davenport1939waring]; Deshouillers, Hennecart, and Landreau have given a model and evidence suggesting a precise density of $0.0999425\ldots$ [deshouillers2006density]. Wooley has shown, unconditionally, that $F_{0}(\mathbb{Z}_{\geq 0}^{3})$ contains $\gg A^{0.91709477}$ integers $a\in[0,A]$ for reals $A\geq 1$ [wooley1995breaking, wooley2000sums, wooley2015sums]. We now recall a result of Hooley [hooley1986HasseWeil, hooley_greaves_harman_huxley_1997] and Heath-Brown [heath1998circle], and state our main result building on it; we then give further details and background.

Theorem (Hooley; Heath-Brown).

For certain Hasse–Weil $L$ -functions, assume automorphy and GRH. Then $\gg_{\epsilon}A^{1-\epsilon}$ integers $a\in[0,A]$ lie in $F_{0}(\mathbb{Z}_{\geq 0}^{3})$ , for any $\epsilon>0$ .

Theorem.

For certain Hasse–Weil $L$ -functions, assume automorphy, GRH, and the Ratios Conjectures. For a certain polynomial $\Delta$ , assume the Square-free Sieve Conjecture. Then $\gg A$ integers $a\in[0,A]$ lie in $F_{0}(\mathbb{Z}_{\geq 0}^{3})$ , and $100\%$ of integers $a\not\equiv\pm 4\bmod{9}$ lie in $F_{0}(\mathbb{Z}^{3})$ .

Both results require estimating sums that roughly take the following form:

(1.1)

\sum_{\bm{c}\in\mathbb{Z}^{6}}\int_{t\in\mathbb{R}}\frac{dt}{\prod_{p\nmid\Delta(\bm{c})}(\textnormal{local $L$-factors})}\cdot(\textnormal{real analysis})\cdot\prod_{p\mid\Delta(\bm{c})}(\textnormal{geometry/$\mathbb{F}_{p}$ + analysis/$\mathbb{Z}_{p}$}).

The utility of GRH in this context has been highlighted by Bombieri; see [bombieri2006riemann]*p. 111. Also, in a function-field setting, [glas2022question] has made [heath1998circle] unconditional, and one could likely simplify our present hypotheses accordingly (since GRH and the Square-free Sieve Conjecture are known over function fields). We only focus on $\mathbb{Q}$ for practical reasons.

For a typical $a$ , the integer solutions to $F_{0}=a$ are expected to be at least exponentially sparse, if they in fact exist. Heath-Brown’s conjecture would imply that the only obstructions to solubility for $F_{0}=a$ , for any $a$ , are local. The naive local-to-global analog of Heath-Brown’s conjecture for $5x^{3}+12y^{3}+9z^{3}$ is known to fail (see e.g. [ghosh2017integral]*p. 691, footnote 3), due to Brauer–Manin obstructions that do not apply to $F_{0}$ [colliot2012groupe]*p. 1304.

We restrict ourselves to a statistical analysis of $F_{0}=a$ over $a\in\mathbb{Z}$ , for $(x,y,z)\in\mathbb{Z}^{3}$ lying in carefully chosen regions, conducted using second moments and the variance framework of [ghosh2017integral, diaconu2019admissible, wang2022thesis, wang2023prime]. This connects naturally to difficult open questions in $6$ variables, e.g. [hooley1986some]*Conjecture 2, which lie beyond the square-root barrier in the classical Hardy–Littlewood circle method. We will attack these questions under standard number-theoretic hypotheses, primarily regarding $L$ -function statistics of Random Matrix Theory (RMT) type. Our work opens with the delta method of [duke1993bounds, heath1996new] (a clean modern form of the Kloosterman method of [kloosterman1926representation], more precise than the upper-bound variant used in [hooley1986HasseWeil, hooley_greaves_harman_huxley_1997]), whose harmonic analysis in principle allows for cancellation over the difficult classical minor arcs. We prove three levels of results, under three levels of hypotheses (the first two levels being relatively soft and qualitative; see Conjectures 1.4 and 1.8). We first recall a general weighted version of Hooley’s conjecture.

Fix a cubic form $F(\bm{x})=F(x_{1},\dots,x_{m})\in\mathbb{Z}[x_{1},\dots,x_{m}]$ in $m\geq 4$ variables with nonzero discriminant. Let $V$ be the hypersurface $F=0$ in $\mathbb{P}^{m-1}_{\mathbb{Q}}$ . Let $\Upsilon$ be the set of $\lfloor m/2\rfloor$ -dimensional vector spaces $L\subseteq\mathbb{Q}^{m}$ over $\mathbb{Q}$ on which $F$ vanishes. On a first reading, we suggest assuming that $m=6$ and $F$ is diagonal, though we will often work generally.

Given a real $X\geq 1$ , and a function $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ , let

(1.2)

N_{F,w}(X)\colonequals\sum_{\bm{x}\in\mathbb{Z}^{m}:\,F(\bm{x})=0}w(\bm{x}/X),\quad N_{F}(X)\colonequals\sum_{\bm{x}\in[-X,X]^{m}:\,F(\bm{x})=0}1;

and if $m=6$ (our main case of interest, in which $\lfloor m/2\rfloor=3$ ), let

(1.3)

E_{F,w}(X)\colonequals N_{F,w}(X)-\mathfrak{S}_{F}\cdot\sigma_{\infty,F,w}\cdot X^{3}-\sum_{L\in\Upsilon}\sum_{\bm{x}\in L\cap\mathbb{Z}^{6}}w(\bm{x}/X),

where $\mathfrak{S}_{F}\ll_{F}1$ is the familiar singular series defined in [wang2023_isolating_special_solutions]*§6, and where

(1.4)

\sigma_{\infty,F,w}\colonequals\lim_{\epsilon\to 0}{(2\epsilon)^{-1}\int_{\lvert F(\bm{x})\rvert\leq\epsilon}d\bm{x}\,w(\bm{x})}\ll_{F,w}1.

One could attribute to Hooley [hooley1986some]*Conjecture 2, Manin (see e.g. [franke1989rational]), Vaughan–Wooley [vaughan1995certain]*Appendix, Peyre [peyre1995hauteurs], et al. the following conjecture:

(1.5)

\lim_{X\to\infty}X^{-3}E_{F,w}(X)=0.

(The original [hooley1986some]*Conjecture 2 for $l=3$ would follow from (1.5) for $F=x_{1}^{3}+\dots+x_{6}^{3}$ , applied to a suitable sequence of weights $w$ .) See [ding2020variance] for another related problem.

Unconditionally, $N_{F}(X)\ll_{\epsilon}X^{7/2}/(\log{X})^{5/2-\epsilon}$ for $X\geq 2$ [vaughan2020some], when $m=6$ and $F$ is diagonal. Under standard hypotheses on the varieties $V_{\bm{c}}\subseteq\mathbb{P}^{m-1}_{\mathbb{Q}}$ cut out by $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ , one can prove the near-optimal Theorem 1.1 for the same $F$ . (Here $\bm{c}\cdot\bm{x}\colonequals\sum_{1\leq i\leq m}c_{i}x_{i}$ .) Let $\Delta(\bm{c})\in\mathbb{Z}[\bm{c}]=\mathbb{Z}[c_{1},\dots,c_{m}]$ be the discriminant polynomial defined in §2. Let

(1.6)

\mathcal{S}_{0}\colonequals\{\bm{c}\in\mathbb{Z}^{m}:\Delta(\bm{c})=0\},\quad\mathcal{S}_{1}\colonequals\{\bm{c}\in\mathbb{Z}^{m}:\Delta(\bm{c})\neq 0\}.

For each $\bm{c}\in\mathcal{S}_{1}$ , one can package local data on $V_{\bm{c}}$ into a Hasse–Weil $L$ -function $L(s,V_{\bm{c}})$ , defined in §3 along with the rest of the list (1.7).

Theorem 1.1 ([hooley1986HasseWeil, hooley_greaves_harman_huxley_1997]; [heath1998circle]).

Assume $m=6$ and $F$ is diagonal. For each $\bm{c}\in\mathcal{S}_{1}$ , assume Conjecture 1.2 for $L(s,V_{\bm{c}})$ . Then $N_{F}(X)\ll_{\epsilon}X^{3+\epsilon}$ for all $\epsilon>0$ .

Conjecture 1.2 concerns automorphy and the Grand Riemann Hypothesis (GRH). Strictly speaking, Hooley and Heath-Brown assume (in their “Hypothesis HW”) GRH plus certain Selberg-type axioms, e.g. analyticity, but it is natural to assume automorphy in place of such axioms. A nice reference bridging these two perspectives is [farmer2019analytic].

The proof of Theorem 1.1 involves GRH on $1/L(s,V_{\bm{c}})$ , surprisingly, coupled with subtle algebro-geometric factors of a different nature. See §2 for details. The $L$ -function ingredients extend directly to general $F$ , but some other ingredients have yet to be generalized.

GRH gives a pointwise upper bound on $1/L(s,V_{\bm{c}})$ for $\operatorname{Re}(s)>1/2$ , sufficient for Theorem 1.1. Going past the critical line (in mean value over $\bm{c}$ ) turns out to require much new work, both with $L$ -functions and with other factors. In terms of $L$ -functions, we mainly use $L(s,V_{\bm{c}})$ for $\operatorname{Re}(s)\geq 1/2-\delta$ , as well as $L(s,V_{\bm{c}},\bigwedge^{2})$ , $\zeta(s)$ , and $L(s,V)$ for $\operatorname{Re}(s)\geq 1-\delta$ . It will be convenient to assume Conjecture 1.2 in full, but average versions might also suffice.

Conjecture 1.2 (HW2).

Let $\bm{c}\in\mathcal{S}_{1}$ . Let $L(s)$ be one of the Hasse–Weil $L$ -functions

(1.7)

L(s,V_{\bm{c}}),\;L(s,V_{\bm{c}},{\textstyle\bigotimes^{2}}),\;L(s,V_{\bm{c}},\operatorname{Sym}^{2}),\;L(s,V_{\bm{c}},{\textstyle\bigwedge^{2}}),\;\zeta(s),\;L(s,V).

Let $L_{v}(s)$ be the local factors (including $L_{\infty}(s)$ , the gamma factor) associated to $L$ .

(1)

There exists an integer $d\geq 1$ , and an isobaric automorphic representation $\Pi$ of $\operatorname{GL}_{d}(\mathbf{A}_{\mathbb{Q}})$ , such that $L_{v}(s)=L_{v}(s,\Pi)$ at all places $v\leq\infty$ .
(2)

$L(s,\Pi)$ has no zeros in the half-plane $\operatorname{Re}(s)>1/2$ .

For the background needed to interpret Conjecture 1.2 (HW2), see §3.

Theorem 1.3.

Suppose $F=x_{1}^{3}+\dots+x_{6}^{3}$ . Assume Conjectures 1.2, 1.4, and 1.5. Then

(1.8)

N_{F}(X)\ll X^{3}.

Let $S\subseteq\mathbb{Z}_{\geq 0}$ . If $S$ has positive lower density in $\mathbb{Z}_{\geq 0}$ , then so does $F_{0}(S^{3})$ .

We use mean-value RMT-type predictions derived from the Moments and Ratios Conjectures of [conrey2005integral, conrey2007applications, conrey2008autocorrelation]. (See also [diaconu2003multiple, vcech2022ratios], and references within, for another important perspective on such conjectures.) The Hasse–Weil $L$ -functions $L(s,V_{\bm{c}})$ form a geometric family, in the sense of [sarnak2016families]. For each $\bm{c}\in\mathcal{S}_{1}$ , define the Dirichlet series

(1.9)

\Phi^{\bm{c},1}(s)\colonequals\zeta(2s)^{-1}L(s+1/2,V)^{-1}L(s,V_{\bm{c}})^{-1}=1/{\zeta(2s)L(s+1/2,V)L(s,V_{\bm{c}})}.

Note that $\zeta(2s)$ and $L(s+1/2,V)$ are independent of $\bm{c}$ .

Conjecture 1.4 (R2’).

Suppose $2\mid m$ . Let $f\colon\mathbb{C}\to\mathbb{C}$ be entire, with $f(s)\ll_{f,b}(1+\lvert\operatorname{Im}(s)\rvert)^{-b}$ on the strip $0\leq\operatorname{Re}(s)\leq 2$ for all $b\in\mathbb{Z}_{\geq 1}$ . Let $Z,N\in\mathbb{R}_{\geq 1}$ with $N\leq Z^{3}$ . If $\sigma_{0}\in(1,2)$ , then

(1.10)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\left\lvert\int_{(\sigma_{0})}ds\,\Phi^{\bm{c},1}(s)\cdot f(s)N^{s}\right\rvert^{2}\ll_{F}Z^{m}N\sup_{0\leq\sigma\leq 2}\int_{\mathbb{R}}dt\,(1+\lvert t\rvert)^{2}\lvert f(\sigma+it)\rvert^{2}.

The contour $(\sigma_{0})$ in (1.10) runs from $s=\sigma_{0}-i\infty$ to $s=\sigma_{0}+i\infty$ . The left-hand side of (1.10) is independent of $\sigma_{0}\in(1,2)$ unconditionally, or further under (HW2). Other versions of Conjecture 1.4 might also suffice for our purposes; for instance, an $\ell^{1+\delta}$ analog of (1.10) (with some nontrivial adjustments) might suffice, the precise norm of $f$ on the right-hand side of (1.10) is not very important, and one might not need to allow such general $f$ .

There are no log factors on the right-hand side of (1.10); the factor $\zeta(2s)^{-1}L(s+1/2,V)^{-1}$ in (1.9), and the integral over $(\sigma_{0})$ in (1.10), play a mollifying role. The statement (R2’) can be derived from (HW2) and the Ratios Conjecture 6.3 (R2 $o$ ); see Proposition 6.8. However, there may well be another route to (R2’) not passing through (R2 $o$ ). The statement (R2’) essentially concerns cancellation in the coefficients of $\Phi^{\bm{c},1}(s)$ over moduli $n$ in a dyadic range; see Proposition 6.10. A similar log-free cancellation statement, [li2022moments]*(1.3), has recently played a crucial role in another context. Furthermore, over function fields (or under (HW2) over $\mathbb{Q}$ ), it should already be possible to obtain partial results towards (R2’), using ideas of [soundararajan2009moments, harper2013sharp, bui2021ratios, florea2021negative, bui2023negative] (after Cauchy–Schwarz over $s$ ).

For our main results, we also need the Square-free Sieve Conjecture (cf. [miller2004one]*p. 956 and [granville1998abc, poonen2003squarefree, bhargava2014geometric]) for the polynomial $\Delta$ , restricted to a certain range $1\leq P\leq Z^{3/2}$ . This hypothesis concerns “unlikely divisors” of the outputs of $\Delta$ . Such hypotheses can be made unconditional over function fields; see e.g. [poonen2003squarefree]*Lemma 7.1.

Conjecture 1.5 (SFSC_p,3).

There exists $\eta_{0}=\eta_{0}(\Delta)\in\mathbb{R}_{>0}$ such that if $Z,P\in\mathbb{R}_{\geq 1}$ and $P\leq Z^{3/2}$ , then $\#\{\bm{c}\in\mathbb{Z}^{m}\cap[-Z,Z]^{m}:\exists\;\textnormal{a prime $p\in[P,2P)$ with $p^{2}\mid\Delta(\bm{c})$}\}\ll_{\Delta}Z^{m}P^{-\eta_{0}}$ .

The Square-free Sieve Conjecture (SFSC_p,3) is used to confront some novel algebro-geometric issues in our work. We use it to prove, for diagonal $F$ , a geometric relative (Conjecture 9.8) of the automorphic Sarnak–Xue Density Hypothesis ([sarnak1991bounds]*Conjecture 1, concerning the extent to which a naive generalization of the Ramanujan Conjecture can fail). This involves sieve-theoretic ideas weighted by somewhat dangerous factors. Even though we do not currently see how to eliminate the use of (SFSC_p,3) over $\mathbb{Q}$ , it is fortunate to be able to reduce Conjecture 9.8 to (SFSC_p,3) when $F$ is diagonal. One can also reduce (SFSC_p,3) for diagonal $F$ to the case $m=4$ (but not to $m=2$ , it seems); see Proposition 9.13.

After Theorem 1.3, our next main result is the following:

Theorem 1.6.

Suppose $m=6$ and $F$ is diagonal. Assume Conjectures 1.2, 1.4, 1.5, and 1.8. Then (1.5) holds for all functions $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ for which

(1.11)

\overline{\{\bm{x}\in\mathbb{R}^{m}:w(\bm{x})\neq 0\}}\subseteq\{\bm{x}\in\mathbb{R}^{m}:x_{1}\cdots x_{m}\neq 0\}.

Therefore, the Hasse principle holds for $V$ . Furthermore, if $F=x_{1}^{3}+\dots+x_{6}^{3}$ , then $100\%$ of integers $a\not\equiv\pm 4\bmod{9}$ lie in $F_{0}(\mathbb{Z}^{3})$ .

For another conditional approach to the Hasse principle for $V$ when $F$ is diagonal, see [swinnerton2001solubility]. Over function fields of characteristic $\geq 7$ , the Hasse principle for $V$ is already known in general when $m=6$ [tian2017hasse]. But our approach has quantitative advantages, which become qualitative when applied to sums of three cubes.

We expect that the condition (1.11) could be removed with enough work. When $F=x_{1}^{3}+\dots+x_{6}^{3}$ , it is in fact possible to do this for free: Theorem 1.6 has the following corollary. (A similar but messier statement is possible for arbitrary diagonal $F$ when $m=6$ .)

Corollary 1.7.

Suppose $F=x_{1}^{3}+\dots+x_{6}^{3}$ . Assume Conjectures 1.2, 1.4, 1.5, and 1.8. Then (1.5) holds for all $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ . Consequently, [hooley1986some]*Conjecture 2 for $l=3$ holds.

Theorem 1.6 makes use of a first-moment estimate for the quantity (1.9) over $\mathcal{S}_{1}$ , and over some mildly localized pieces of $\mathcal{S}_{1}$ (“adelic perturbations” of $\mathcal{S}_{1}$ restricted by a parameter $M$ ). For $\bm{b}=(b_{1},\dots,b_{m})\in\mathbb{Z}^{m}$ and $M\in\mathbb{R}_{>0}$ , consider the box

(1.12)

\mathcal{B}_{M}(\bm{b})\colonequals[b_{1}/M,(b_{1}+1)/M)\times\dots\times[b_{m}/M,(b_{m}+1)/M)\subseteq\mathbb{R}^{m}.

For $Z\in\mathbb{R}_{>0}$ , let $Z\cdot\mathcal{B}_{M}(\bm{b})$ denote the dilate $\{Z\bm{r}:\bm{r}\in\mathcal{B}_{M}(\bm{b})\}\subseteq\mathbb{R}^{m}$ . For each $Z\in\mathbb{R}_{\geq 2}$ , let

(1.13)

\sigma(Z)\colonequals 1/2+1/\log{Z}.

Conjecture 1.8 (RA1 $o$ ).

Suppose $2\mid m$ , and assume Conjecture 1.2. Let $M\in\mathbb{R}_{\geq 1}$ ; let $n_{0}\in\mathbb{Z}\cap[1,M]$ and $\bm{a},\bm{b}\in\mathbb{Z}^{m}\cap[-M,M]^{m}$ . Let $A_{F,1}^{\bm{a},n_{0}}(s)$ be defined as in §6.3.1 (in terms of $F$ , $\bm{a}$ , $n_{0}$ ). If $Z\in\mathbb{R}_{\geq 2}$ and $t\in[-\log{Z},\log{Z}]$ , then for $s=\sigma(Z)+it$ , we have

(1.14)

\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}\Phi^{\bm{c},1}(s)=\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}(1+o_{F,M;Z\to\infty}(1))\cdot A_{F,1}^{\bm{a},n_{0}}(s).

Here $A_{F,1}^{\bm{a},n_{0}}(s)$ is a Dirichlet series with an Euler product, absolutely convergent on the half-plane $\operatorname{Re}(s)>1/3$ . The terms $o_{F,M;Z\to\infty}(1)$ are required to tend to $0$ as $Z\to\infty$ (when $F$ , $M$ are fixed). See Proposition 6.13 for the main use of Conjecture 1.8.

The Ratios Conjectures include (RA1 $o$ ), even with a power saving; see §6.3.1 for details. The choice (1.13) is permissible according to [conrey2007applications]*(2.11b). The essential feature of (1.13) for us is that $(\sigma(Z)-\frac{1}{2})\cdot\log{Z}$ is positive and independent of $Z$ (but its precise constant value is not important). We could get away with a larger choice of $\sigma(Z)$ if we assumed a correspondingly stronger error term in (1.14); but the present formulation of (RA1 $o$ ) is clean, and easy to compare with other literature.¹¹1Note that if $s=\sigma(Z)$ , then in (1.9), we have $\zeta(2s)\asymp\log{Z}$ and $L(s+1/2,V)\asymp(\log{Z})^{r_{F}}$ , where $r_{F}\in\mathbb{Z}_{\geq 0}$ is given explicitly for diagonal $F$ by [wang2022thesis]*Lemma 8.6.7. The moments of $1/L$ we consider (over our orthogonal family of $L$ -functions $L(s,V_{\bm{c}}$ )) are thus analogous to central moments over symplectic families. The paper [florea2021negative] seems to come close to proving (RA1 $o$ ) for a different family of $L$ -functions, over a function field.

We believe (RA1 $o$ ), like (R2’), represents a tantalizing research direction. There is another direction worth mentioning. In light of the log-free square-root cancellation in $\ell^{2}$ conjectured in (1.10), one may hope that “better than square-root cancellation” occurs in $\ell^{2-\epsilon}$ , by analogy with [harper2023typical]*(1.2) (an attractive conjecture based on random multiplicative functions and multiplicative chaos; see e.g. [gorodetsky2021magic, harper2023typical] for details and references). If true, this would provide additional cancellation in Proposition 7.5 (one of the key ingredients for Theorem 1.3), and thus provide an alternative approach to Theorem 1.6 (but not to Theorem 1.9).

Our final main result, Theorem 1.9, goes beyond Theorem 1.6.

Theorem 1.9.

Suppose $m=6$ and $F$ is diagonal. Assume Conjectures 1.2, 1.5 with $\eta_{0}$ , 1.10 with $\eta_{1}$ , and 1.11 with $H$ . Then there exists a real $\delta=\delta(\eta_{0},\eta_{1},\deg{H})>0$ such that $E_{F,w}(X)\ll_{F,w}X^{3-\delta}$ holds for all functions $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ satisfying (1.11).

Our methods would allow one to prove a version of Theorem 1.9 uniform over small archimedean and non-archimedean perturbations to $E_{F,w}(X)$ . By [wang2023prime]*Theorem 1.2, one could then show under the hypotheses of Theorem 1.9 that if $F=x_{1}^{3}+\dots+x_{6}^{3}$ , then $100\%$ of primes $p\not\equiv\pm 4\bmod{9}$ lie in $F_{0}(\mathbb{Z}^{3})$ . One would also be able to give a power-saving analog of Corollary 1.7. But to give full details would obscure our exposition.

The power saving in Theorem 1.9 is small and complicated. (Egregious “exponent divisions” occur in Lemma 7.13 and Proposition 9.5, due to our use of (4.3); and similarly in (10.8), due to (4.4).) It would be very interesting to understand the limits of what one can hope for.

Conjecture 1.10 (RA1 $\delta$ ).

Suppose $2\mid m$ , and assume Conjecture 1.2. Let $Z\in\mathbb{R}_{\geq 2}$ and $M\in[1,Z^{\eta_{1}}]$ . Let $n_{0}\in\mathbb{Z}\cap[1,M]$ and $\bm{a},\bm{b}\in\mathbb{Z}^{m}\cap[-M,M]^{m}$ . There exists a real $\eta_{1}=\eta_{1}(F)>0$ , depending only on $F$ , such that if $t\in[-M,M]$ and $s=\sigma(Z)+it$ , then

(1.15)

\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}\Phi^{\bm{c},1}(s)=\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}(1+O_{F}(Z^{-\eta_{1}}))\cdot A_{F,1}^{\bm{a},n_{0}}(s).

For Theorem 1.9, certain degenerate residue classes play a larger role in local calculations than for Theorem 1.6. To pacify these residue classes, we need effective control on the variation of an individual local factor $L_{p}(s,V_{\bm{c}})$ over $\bm{c}\in\mathcal{S}_{1}$ . We work $p$ -adically for convenience, using Remark 3.1 to define $L_{p}(s,V_{\bm{c}})$ for each $\bm{c}\in\mathbb{Z}_{p}^{m}$ with $\Delta(\bm{c})\neq 0$ .

Conjecture 1.11 (EKL).

There exists a nonzero homogeneous polynomial $H\in\mathbb{Z}[\bm{c}]$ , with $H/\Delta\in\mathbb{Z}[\bm{c}]$ , such that for all primes $p$ and tuples $\bm{a},\bm{b}\in\mathbb{Z}_{p}^{m}$ with $H(\bm{b})\neq 0$ and $\bm{a}\equiv\bm{b}\bmod{pH(\bm{b})}$ , we have $H(\bm{a})\neq 0$ and $L_{p}(s,V_{\bm{a}})=L_{p}(s,V_{\bm{b}})$ .

Conjecture 1.11 is an effective Krasner-type statement for $L_{p}(s,V_{\bm{c}})$ . A soft version follows from [kisin1999local], and suffices for Theorem 1.6 but not for Theorem 1.9. When $m=4$ , it should be possible to prove Conjecture 1.11 (with $H\in\mathbb{Q}^{\times}\cdot\Delta$ ) using a minimal model for the Jacobian of $V_{\bm{c}}$ . In general, one might hope to take $H$ to be a power of $\Delta$ (or perhaps $\Delta$ itself).

1.1. Proof overview

§2 gives background on discriminants and the delta method. The delta method (see (2.10)) connects the point count $N_{F,w}(X)$ (from (1.2)) to the local behavior of the intersections $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ over $\mathbb{F}_{p}$ , $\mathbb{Z}_{p}$ , $\mathbb{R}$ , and other rings, as $\bm{c}\in\mathbb{Z}^{m}$ and $p$ vary. Cf. (1.1). We highlight several distinct sources of epsilon in the Hooley–Heath-Brown Theorem 1.1, and state a result from [wang2023_isolating_special_solutions] (over $\mathcal{S}_{0}$ ) addressing one such source.

§3 provides background on Hasse–Weil $L$ -functions and automorphic $L$ -functions.

§4 gives some local control on polynomials and $L$ -factors (based in part on [kisin1999local]); we need this for some local estimates and calculations.

§5 gives a useful “smooth framework” for dyadic decomposition and separation of variables. This lets us break certain key sums throughout the paper into more manageable pieces.

§6 derives some $L$ -function statistics over $\bm{c}\in\mathcal{S}_{1}$ , after first doing local calculations (in the spirit of the Deligne–Katz equidistribution theorem) connected to RMT Symmetry Types (cf. [sarnak2016families]*Universality Conjecture). In particular, we state, and build on, some cases of the Ratios Conjectures for $L(s,V_{\bm{c}})$ . Importantly here, the Ratios Recipe (see §6.3) can only apply once we restrict to $\bm{c}\in\mathcal{S}_{1}$ . The recipe, naively extended to all $\bm{c}\in\mathbb{Z}^{m}$ , would give false results (failing to detect the special subvarieties $L\in\Upsilon$ on $F=0$ isolated in [wang2023_isolating_special_solutions]).

§7 begins to connect the “pure” $L$ -function statistics from §6 to the delta method. We approximate certain Dirichlet series “past” the critical line, in a reasonably simple and uniform way over $\bm{c}\in\mathcal{S}_{1}$ , and control the resulting “approximation errors” on average. Handling these “errors” demands careful use of Hölder and other ideas. For example, by algorithmic tree-like means, we construct in §7.3 a small exceptional set away from which one may apply the “pure” Conjectures 1.8 and 1.10; it is also here that Conjecture 1.11 plays some key role.

The “mollified” series $\Phi^{\bm{c},1}(s)$ from (1.9) not only makes the formulas in §6 nicer, but (as we will explain in §7) also holds significance in (2.10); this double significance, though innocent at first glance, secretly reflects a randomness property (connected to Deligne–Katz) stemming from the fact that $\deg{F}\geq 3$ . Throughout the paper, we take much advantage of the structure of (1.9); this is essential in Conjecture 1.4 and Lemma 6.15, for instance.

§8 proves new integral bounds sensitive to some real geometry (involving discriminants). Our approach shares some important features with [huang2020density] (a beautiful recent paper on approximate integral points). In addition, we have several parameters of interest, and must obtain genuinely multivariate decay. Keeping track of uniformity is tricky.

§9, like §8, proves some new “discriminating” pointwise estimates, but on complete exponential sums instead of oscillatory integrals. We then apply these to formulate and address (under Conjecture 1.5) a geometric analog of the Sarnak–Xue Density Hypothesis. Finite-field geometry (see [wang2023dichotomous]) and the geometric sieve (see [ekedahl1991infinite, bhargava2014geometric]) both play an important role here, as does a nice result of Busé and Jouanolou on discriminants (see Theorem 2.2). It is crucial throughout §9 that $\deg{F}\geq 3$ (see e.g. Remark 9.2); this reflects a “randomness” not present for quadrics.

§10 ties everything together to prove our main results. We also isolate “axioms” that—if true—would allow for non-diagonal $F$ ; see Theorems 10.5, 10.7, and 10.8. Here we only consider $\bm{c}\in\mathcal{S}_{1}$ ; there are also separate issues for $\bm{c}\in\mathcal{S}_{0}$ (see [wang2023_isolating_special_solutions]*Remark 1.6).

For Theorem 1.3, see §10.2. Here we use an “entirely positive” Hölder argument over $\bm{c}\in\mathcal{S}_{1}$ : we do not detect any cancellation over $\bm{c}$ that would go beyond a log-free “Mertens-type heuristic on average” (cf. [ng2004distribution]*Theorem 1(iii)) over $\bm{c}$ . Despite the “decoupling” convenience and power of Hölder, we must therefore be careful in §10.2 to obtain $\epsilon$ -free bounds. (The structure of §10.2 is inspired by our work with the large sieve in [wang2023_large_sieve_diagonal_cubic_forms].)

The proof of Theorem 1.3 tells us (conditionally) that even if one takes absolute values over $\bm{c}\in\mathcal{S}_{1}$ in (2.10), the $X^{\epsilon}$ allowance of [hooley_greaves_harman_huxley_1997, heath1998circle] is unnecessary; see (10.14) in Theorem 10.5. Theorem 1.3 also highlights a nontrivial use of the log-free order of magnitude in Conjecture 1.4, a robust qualitative prediction that (if true) could perhaps be explained in other ways (not just following the rather arithmetic Ratios Recipe).

For Theorem 1.6, Corollary 1.7, and Theorem 1.9, see §10.3. In most ranges, the “entirely positive” moment estimates of §10.2 still suffice. But this time, in a few key ranges, we identify cancellation over $\bm{c}\in\mathcal{S}_{1}$ (via §7). One critical step here is a reduction, via §8, to large moduli in (2.10) (over $\bm{c}\in\mathcal{S}_{1}$ ), over which certain “mollified” RMT-type main terms vanish (cf. Lemma 6.15). There is also an alternative approach to cancellation (which we do not pursue): instead of Lemma 6.15, we could use the fact that $\int_{\mathbb{R}^{m}}d\bm{c}\,J_{\bm{c},X}(n)=0$ (provided $w(\bm{0})=0$ ), where $J_{\bm{c},X}(n)$ is defined as in (2.9); cf. [wang2021_HLH_vs_RMT]*Observation 10.7.

1.2. Conventions

We write $f\ll_{S}g$ , or $g\gg_{S}f$ , to mean $\lvert f\rvert\leq Cg$ for some $C=C(S)>0$ . We let $O_{S}(g)$ denote a quantity that is $\ll_{S}g$ . We write $f\asymp_{S}g$ if $f\ll_{S}g\ll_{S}f$ . We let $o_{S;X\to\infty}(g)$ denote a quantity $f$ such that for every $\epsilon>0$ , there exists $X_{0}=X_{0}(\epsilon,S)>0$ such that $\lvert f\rvert\leq\epsilon g$ holds for all $X\geq X_{0}$ . When making estimates, we think of $m$ , $F$ , $w$ as fixed, but may still occasionally write $\ll_{F}$ (or similar) for emphasis.

We frequently use indicator notation, letting $\bm{1}_{E}\colonequals 1$ if $E$ holds, and $\bm{1}_{E}\colonequals 0$ otherwise. For any nonempty set $S$ with an obvious measure (e.g. the counting measure on a finite set, or the usual Haar measure on $\mathbb{Z}_{p}^{m}$ ), we let $\mathbb{E}_{b\in S}[f(b)]$ denote the average of $f(b)$ over $b\in S$ .

We let $\mathbb{Z}_{\geq 0}\colonequals\{a\in\mathbb{Z}:a\geq 0\}$ , and similarly define sets like $\mathbb{Z}_{\neq 0}$ , $\mathbb{R}_{>1}$ , $\mathbb{R}_{\geq 2}$ . For $c\in\mathbb{Z}_{\neq 0}$ , we let $v_{p}(c)$ denote the $p$ -adic valuation of $c$ . For $n\in\mathbb{Z}_{\geq 1}$ , we let $\varphi(n)$ denote the totient function, $\omega(n)$ the number of distinct prime factors of $n$ , and $\operatorname{rad}(n)$ the radical of $n$ .

We let $C^{\infty}_{c}(\mathbb{R}^{s})$ (resp. $C^{\infty}_{c}(\mathbb{R}^{s})\otimes\mathbb{C}$ ) denote the set of smooth compactly supported functions $\mathbb{R}^{s}\to\mathbb{R}$ (resp. $\mathbb{R}^{s}\to\mathbb{C}$ ). For any function $f=f(\bm{u})$ , we let $\operatorname{Supp}{f}$ denote the closure of $\{\bm{u}:f(\bm{u})\neq 0\}$ in the domain of $f$ ; so for instance, the left-hand side of (1.11) equals $\operatorname{Supp}{w}$ .

We let $e(t)\colonequals e^{2\pi it}$ , and $e_{r}(t)\colonequals e(t/r)$ . In integrals, we use notation analogous to summation notation. For instance, we write $\int_{X}dx\,f(x)$ to mean $\int_{X}f(x)\,dx$ (in conventional notation), and we then write $\int_{X\times Y}dx\,dy\,f(x,y)$ to mean $\int_{X}dx\,(\int_{Y}dy\,f(x,y))$ .

We need concise notation for $L^{p}$ -norms and $\ell^{p}$ -norms. If $f$ is a quantity depending on a scalar or vector variable $t$ (and possibly also on other variables), we write $\lVert f\rVert_{L^{p}_{t}(S)}\colonequals(\int_{t\in S}dt\,\lvert f\rvert^{p})^{1/p}$ or $\lVert f\rVert_{\ell^{p}_{t}(S)}\colonequals(\sum_{t\in S}\lvert f\rvert^{p})^{1/p}$ to denote the $p$ -norm of $f$ over $t\in S$ , according as $t$ is a continuous or discrete variable, respectively. If the variable $t$ is clear from context, we may omit it.

We let $\partial_{u}\colonequals\partial/\partial u$ for $u\in\mathbb{R}$ . When doing calculus in $d$ dimensions, a multi-index is a tuple of $d$ nonnegative integers (where $d$ will always be clear from context). Given a multi-index $\bm{\alpha}\geq 0$ , we let $\lvert\bm{\alpha}\rvert$ denote the sum of the coordinates of $\bm{\alpha}$ . For a vector $\bm{u}\in\mathbb{R}^{s}$ , we let $\lVert\bm{u}\rVert\colonequals\max_{i}(\lvert u_{i}\rvert)$ , and write $d\bm{u}\colonequals du_{1}\cdots du_{s}$ and $\partial_{\bm{u}}^{\bm{\alpha}}\colonequals\partial_{u_{1}}^{\alpha_{1}}\cdots\partial_{u_{s}}^{\alpha_{s}}$ . For example, if $f=f(\bm{u})\in C^{\infty}_{c}(\mathbb{R}^{s})\otimes\mathbb{C}$ and $k\in\mathbb{Z}_{\geq 0}$ , then under our conventions,

\max_{\lvert\bm{\alpha}\rvert\leq k}\max_{\bm{u}\in\mathbb{R}^{s}}{\lvert\partial_{\bm{u}}^{\bm{\alpha}}{f}\rvert}=\max_{\alpha_{1},\dots,\alpha_{s}\geq 0:\,\alpha_{1}+\dots+\alpha_{s}\leq k}\,\max_{u_{1},\dots,u_{s}\in\mathbb{R}}{\lvert\partial_{u_{1}}^{\alpha_{1}}\cdots\partial_{u_{s}}^{\alpha_{s}}f\rvert}.

For (repeated) later use, we fix a function $\nu_{0}\in C^{\infty}_{c}(\mathbb{R}^{m})$ , supported on $[-2,2]^{m}$ , with $0\leq\nu_{0}\leq 1$ everywhere, $\nu_{0}=1$ on $[-1/2,1/2]^{m}$ , and $\int_{\mathbb{R}^{m}}d\bm{x}\,\nu_{0}(\bm{x})=1$ . We also fix a radial²²2in the Euclidean sense (so that the value of $\nu_{1}(\bm{x})$ depends only on $x_{1}^{2}+\dots+x_{m}^{2}$ ) function $\nu_{1}\in C^{\infty}_{c}(\mathbb{R}^{m})$ , supported on the annulus $m\leq x_{1}^{2}+\dots+x_{m}^{2}\leq m^{2}$ (so that $\nu_{1}|_{[-1,1]^{m}}=0$ and $\operatorname{Supp}{\nu_{1}}\subseteq[-m,m]^{m}$ ), such that $\int_{\lambda>0}\frac{d\lambda}{\lambda}\,\nu_{1}(\bm{x}/\lambda)=1$ for all $\bm{x}\in\mathbb{R}^{m}\setminus\{\bm{0}\}$ .

2. Background on discriminants and delta

Let $\operatorname{disc}(F)\neq 0$ be the discriminant of $F$ . Let $m_{\ast}\colonequals m-3$ . Define the bi-homogeneous polynomial expression $\operatorname{disc}(F,\bm{c})$ as in [wang2023dichotomous]*§3, so that if $\bm{c}\in\mathbb{C}^{m}$ , then $\operatorname{disc}(F,\bm{c})\neq 0$ if and only if the variety $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ in $\mathbb{P}^{m-1}_{\mathbb{C}}$ is smooth of dimension $m_{\ast}$ . Let

(2.1)

\Delta(\bm{c})\colonequals\operatorname{disc}(F)\cdot\operatorname{disc}(F,\bm{c});

then for any $\bm{c}\in\mathbb{Z}^{m}$ and prime $p$ with $p\nmid\Delta(\bm{c})$ , the varieties $F(\bm{x})=0$ and $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ in $\mathbb{P}^{m-1}_{\mathbb{F}_{p}}$ are both smooth. Now recall $\mathcal{S}_{0}$ , $\mathcal{S}_{1}$ from (1.6). For each $\bm{c}\in\mathcal{S}_{1}$ , let

(2.2)

\mathcal{N}^{\bm{c}}\colonequals\{n\geq 1:p\mid n\Rightarrow p\nmid\Delta(\bm{c})\},\quad\mathcal{N}_{\bm{c}}\colonequals\{n\geq 1:p\mid n\Rightarrow p\mid\Delta(\bm{c})\}.

Lemma 2.1.

Let $N,R\geq 1$ be integers. Then $\lvert\{N\leq n<2N:n\mid R^{\infty}\}\rvert\ll_{\epsilon}(RN)^{\epsilon}$ , where $n\mid R^{\infty}$ means $\operatorname{rad}(n)\mid R$ . In particular, if $\bm{c}\in\mathcal{S}_{1}$ , then $\lvert\{N\leq n<2N:n\in\mathcal{N}_{\bm{c}}\}\rvert\ll_{\epsilon}\lVert\bm{c}\rVert^{\epsilon}N^{\epsilon}$ .

Proof.

See e.g. [heath1998circle]*antepenultimate display of p. 683. ∎

As usual, we write $\nabla{F}=(\partial F/\partial x_{1},\dots,\partial F/\partial x_{m})$ . It is known that the equation $\Delta(\bm{c})=0$ defines the projective dual variety of $V$ (see e.g. [wang2023dichotomous]*Proposition 4.4). So

(2.3)

\Delta(\nabla{F}(\bm{x}))/F(\bm{x})\in\mathbb{Q}[\bm{x}].

Also, $\Delta(\bm{c})$ is irreducible in $\overline{\mathbb{Q}}[c_{1},\dots,c_{m}]$ , and has total degree $\deg{\Delta}=3\cdot 2^{m-2}$ in $c_{1},\dots,c_{m}$ . In particular, $\deg\Delta\geq 12$ (since $m\geq 4$ ), so by [castryck2020dimension]*Theorem 1, we have

(2.4)

\lvert\mathcal{S}_{0}\cap[-Z,Z]^{m}\rvert\ll_{m}Z^{m-2}

for all reals $Z\geq 1$ . (For diagonal $F$ , more is known, e.g. $\lvert\mathcal{S}_{0}\cap[-Z,Z]^{m}\rvert\ll_{m,\epsilon}Z^{m/2+\epsilon}$ .)

One can express $\operatorname{disc}(F,\bm{c})$ in terms of the discriminant of a cubic form in $m-1$ variables:

(2.5)

\operatorname{disc}(F,\bm{c})=\pm\operatorname{disc}(F(x_{1},\dots,x_{m-1},-(c_{1}x_{1}+\dots+c_{m-1}x_{m-1})/c_{m}))\cdot c_{m}^{3\cdot 2^{m-2}}

[wang2023dichotomous]*Proposition 3.2. This lets us bring into play a nice result of [buse2014discriminant] (though the weaker classical result [hooley1988nonary]*(84) on p. 62 would also suffice for most of our needs):

Theorem 2.2 ([buse2014discriminant]*Corollary 4.30).

Let $R$ be a ring and let $f\in R[x_{1},\dots,x_{m-1}]$ be a homogeneous polynomial. If $1\leq a\leq m-1$ , then there exists $N\geq 0$ such that $x_{a}^{N}\operatorname{disc}(f)$ lies in the homogeneous ideal of $R[x_{1},\dots,x_{m-1}]$ generated by

f,\;\{(\partial_{x_{i}}{f})(\partial_{x_{j}}{f}):1\leq i,j\leq m-1\}.

Via (2.5), Theorem 2.2 has the following corollary, useful in §9:

Corollary 2.3.

If $1\leq a,b\leq m$ , then there exists $N\geq 0$ such that $x_{a}^{N}c_{b}^{N}\Delta(\bm{c})$ lies in the homogeneous ideal of $\mathbb{Z}[x_{1},\dots,x_{m},c_{1},\dots,c_{m}]$ generated by

F(\bm{x}),\;\bm{c}\cdot\bm{x},\;\{(c_{k}\cdot\partial_{x_{i}}{F}-c_{i}\cdot\partial_{x_{k}}{F})(c_{k}\cdot\partial_{x_{j}}{F}-c_{j}\cdot\partial_{x_{k}}{F}):1\leq i,j,k\leq m\}.

Proof.

Assume $b=m$ . Then use (2.1), (2.5), Theorem 2.2 with $R=\mathbb{Z}[c_{1}/c_{m},\dots,c_{m-1}/c_{m}]$ , and the congruence $-(c_{1}x_{1}+\dots+c_{m-1}x_{m-1})/c_{m}\equiv x_{m}\bmod{(\bm{c}/c_{m})\cdot\bm{x}}$ in $R[x_{1},\dots,x_{m}]$ . ∎

Fix $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ satisfying (1.11) if $F$ is diagonal, or satisfying

(2.6)

\operatorname{Supp}{w}\subseteq\{\bm{x}\in\mathbb{R}^{m}:\det(\operatorname{Hess}{F}(\bm{x}))\neq 0\}

in general.³³3For convenience, we will maintain this hypothesis for the rest of the paper, except when specified otherwise. Recall $N_{F,w}(X)$ from (1.2). The delta method of [duke1993bounds, heath1996new] allows one to express $N_{F,w}(X)$ , up to a negligible error, as a sum over $\bm{c}\in\mathbb{Z}^{m}$ of “adelic” data. We use the precise setup from [wang2023_isolating_special_solutions]*§1 (based on [duke1993bounds, heath1996new, heath1998circle]).

Let $h\colon(0,\infty)\times\mathbb{R}\to\mathbb{R}$ be the smooth function given by [heath1998circle]*(2.3); we will need the full definition later, in §8. For each real $X\geq 1$ , let $Y\colonequals X^{3/2}$ . Let

(2.7)	$\displaystyle I_{\bm{c}}(n)$	$\displaystyle\colonequals\int_{\bm{x}\in\mathbb{R}^{m}}d\bm{x}\,w(\bm{x}/X)h(n/Y,F(\bm{x})/Y^{2})e(-\bm{c}\cdot\bm{x}/n),$
(2.8)	$\displaystyle S_{\bm{c}}(n)$	$\displaystyle\colonequals\sum_{1\leq a\leq n:\,\gcd(a,n)=1}\,\sum_{1\leq\bm{x}\leq n}e_{n}(aF(\bm{x})+\bm{c}\cdot\bm{x}),$
(2.9)	$\displaystyle S^{\natural}_{\bm{c}}(n)$	$\displaystyle\colonequals n^{-(m+1)/2}S_{\bm{c}}(n),\quad J_{\bm{c},X}(n)\colonequals X^{-m}I_{\bm{c}}(n).$

A simple rearrangement of [wang2023_isolating_special_solutions]*(1.3) gives (for all $A>0$ )

(2.10)

(1+O_{A}(Y^{-A}))\cdot N_{F,w}(X)/X^{m-3}=\sum_{n\geq 1}\sum_{\bm{c}\in\mathbb{Z}^{m}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n).

The infinite sums in (2.10) are essentially finite, by the following standard result:

Proposition 2.4.

For some constant $A_{0}=A_{0}(F,w)>0$ , we have $J_{\bm{c},X}(n)=0$ for all $n\geq A_{0}Y$ . Also, $J_{\bm{c},X}(n)\ll_{\epsilon,A}X^{-A}$ holds whenever $\epsilon,A>0$ and $\lVert\bm{c}\rVert\geq X^{1/2+\epsilon}$ .

Proof.

See e.g. [wang2023_isolating_special_solutions]*Proposition 5.1. ∎

We now recall some background (cf. [hooley1986HasseWeil]*§§5–6) on the sums $S_{\bm{c}}(n)$ . It is known that $S_{\bm{c}}(n)$ , and thus $S^{\natural}_{\bm{c}}(n)$ too, is multiplicative in $n$ . The Dirichlet series

(2.11)

\Phi(\bm{c},s)\colonequals\sum_{n\geq 1}n^{-s}S^{\natural}_{\bm{c}}(n)

thus has an Euler product. At prime powers, $S_{\bm{c}}$ is related to certain point counts in projective space. Given a prime power $q$ , let $\mathcal{V}(\mathbb{F}_{q})$ be the set of $\mathbb{F}_{q}$ -points on the variety $F(\bm{x})=0$ in $\mathbb{P}^{m-1}_{\mathbb{F}_{q}}$ , let $\mathcal{V}_{\bm{c}}(\mathbb{F}_{q})$ be the set of $\mathbb{F}_{q}$ -points on the variety $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ in $\mathbb{P}^{m-1}_{\mathbb{F}_{q}}$ , and let

E_{F}(q)\colonequals\lvert\mathcal{V}(\mathbb{F}_{q})\rvert-\lvert\mathbb{P}^{1+m_{\ast}}(\mathbb{F}_{q})\rvert,\quad E_{\bm{c}}(q)\colonequals\lvert\mathcal{V}_{\bm{c}}(\mathbb{F}_{q})\rvert-\lvert\mathbb{P}^{m_{\ast}}(\mathbb{F}_{q})\rvert,

where $\lvert\mathbb{P}^{d}(\mathbb{F}_{q})\rvert=(q^{d+1}-1)/(q-1)$ . Then let

(2.12)

E^{\natural}_{F}(q)\colonequals q^{-(1+m_{\ast})/2}E_{F}(q),\quad E^{\natural}_{\bm{c}}(q)\colonequals q^{-m_{\ast}/2}E_{\bm{c}}(q).

If $p\nmid\bm{c}$ (e.g. if $p\nmid\Delta(\bm{c})$ ), then by [wang2022thesis]*Proposition 3.2.4, we have

(2.13)

S^{\natural}_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p)-p^{-1/2}E^{\natural}_{F}(p).

If $p\nmid\Delta(\bm{c})$ and $l\geq 2$ , then by [wang2022thesis]*Proposition 3.2.6, we have

(2.14)

S^{\natural}_{\bm{c}}(p^{l})=0.

Recall $\mathcal{S}_{0}$ , $\mathcal{S}_{1}$ from (1.6). For any set $\mathcal{S}\subseteq\mathbb{Z}^{m}$ , let

(2.15)

\Sigma^{\natural}(X,\mathcal{S})\colonequals\sum_{n\geq 1}\sum_{\bm{c}\in\mathcal{S}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n),\quad\Sigma(X,\mathcal{S})\colonequals X^{m-3}\Sigma^{\natural}(X,\mathcal{S}).

If $\bm{c}\in\mathcal{S}_{0}$ , then $\Phi(\bm{c},s)$ can resemble $\zeta(s-1/2)$ (in some sense), leading to the following result:

Theorem 2.5 ([wang2023_isolating_special_solutions]).

Suppose $m=6$ and $F$ is diagonal. Then

(2.16)

\Sigma(X,\mathcal{S}_{0})=O_{F,w,\epsilon}(X^{2.75+\epsilon})+\mathfrak{S}_{F}\cdot\sigma_{\infty,F,w}\cdot X^{3}+\sum_{L\in\Upsilon}\sum_{\bm{x}\in L\cap\mathbb{Z}^{6}}w(\bm{x}/X),

unconditionally. In particular, $\Sigma(X,\mathcal{S}_{0})\ll_{F,w}X^{3}$ .

Proof.

This follows from [wang2023_isolating_special_solutions]*Corollary 1.2 and (1.7), even if we relax the condition (2.6) to $\bm{0}\notin\operatorname{Supp}{w}$ . The earlier paper [heath1998circle] had proved $\Sigma(X,\mathcal{S}_{0})\ll_{F,w,\epsilon}X^{3+\epsilon}$ . ∎

Since $N_{F,w}(X)=\Sigma(X,\mathcal{S}_{0})+\Sigma(X,\mathcal{S}_{1})+O_{A}(X^{-A})$ , we may thus concentrate on $\Sigma(X,\mathcal{S}_{1})$ . The rest of §2 provides some technical context for our work (in comparison with Theorem 1.1 due to [hooley_greaves_harman_huxley_1997, heath1998circle]), but is not logically necessary for the paper.

If $\bm{c}\in\mathcal{S}_{1}$ , then by (2.13), (2.14), and (3.4), one might expect $\Phi(\bm{c},s)$ to resemble $L(s,V_{\bm{c}})^{(-1)^{m_{\ast}}}$ , up to a factor absolutely convergent for $\operatorname{Re}(s)>1/2$ ; cf. [wang2023_large_sieve_diagonal_cubic_forms]*(2.4). With this intuition in mind, let us now recall how one can prove (as is key for Theorem 1.1)

(2.17)

\Sigma(X,\mathcal{S}_{1})\ll_{\epsilon}X^{3(m-2)/4+\epsilon},\quad\textnormal{or equivalently }\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{\epsilon}X^{(6-m)/4+\epsilon},

under Conjecture 1.2 (HW2) for $L(s,V_{\bm{c}})$ , when $m$ is even and $F$ is diagonal. We find it illuminating to work in this generality, but key here is that $3(m-2)/4=3$ when $m=6$ .

Conditional proof sketch for (2.17).

For this proof sketch only, let $Z\colonequals X^{1/2+\epsilon_{0}}$ . Let

(2.18)

\Psi^{\bm{c},1}(s)\colonequals 1/L(s,V_{\bm{c}}),\quad\Psi^{\bm{c},2}(s)\colonequals\Phi(\bm{c},s)L(s,V_{\bm{c}}),

for $\bm{c}\in\mathcal{S}_{1}$ ; then $\Phi=\Psi^{\bm{c},1}\Psi^{\bm{c},2}$ . Let $\mu_{\bm{c}}(n)$ , $a^{!}_{\bm{c}}(n)$ be the $n$ th coefficients of the Dirichlet series $\Psi^{\bm{c},1}$ , $\Psi^{\bm{c},2}$ , respectively. Then the following hold (see e.g. [wang2023_large_sieve_diagonal_cubic_forms]*Proposition 4.12):

]
[B1’

For $N\leq Z^{3}$ , we have $\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\sum_{N\leq n<2N}\lvert a^{!}_{\bm{c}}(n)\rvert\ll_{\epsilon}Z^{m+\epsilon}N^{1/2}$ .
[B2’

For $N\leq Z^{3}$ , we have $\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}(\sum_{N\leq n<2N}\lvert a^{!}_{\bm{c}}(n)\rvert)^{2}\ll_{\epsilon}Z^{m+\epsilon}N$ .

The arguments in [hooley1986HasseWeil, hooley_greaves_harman_huxley_1997] and [heath1998circle] can then loosely be interpreted as

(H1)

using partial summation over $n\in[N,2N)$ to “factor out” $J_{\bm{c},X}(n)$ from the sum over $n$ in (2.15) (for $\mathcal{S}=\mathcal{S}_{1}$ ), and then bounding the $J$ -contribution in $\ell^{\infty}_{n}([N,2N))$ ;
(H2)

expanding $S^{\natural}_{\bm{c}}=\mu_{\bm{c}}\ast a^{!}_{\bm{c}}$ using $\Phi=\Psi^{\bm{c},1}\Psi^{\bm{c},2}$ ;
(H3)

using GRH to bound the $\Psi^{\bm{c},1}$ -contribution in $\ell^{\infty}_{\bm{c}}(\mathcal{S}_{1}\cap[-Z,Z]^{m})$ ; and
(H4)

using [B1’] afterwards, to bound the $\Psi^{\bm{c},2}$ -contribution in $\ell^{1}_{\bm{c}}(\mathcal{S}_{1}\cap[-Z,Z]^{m})$ .

Upon dyadic summation over $1\ll N\ll Y$ , one gets $\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{\epsilon_{0}}X^{(6-m)/4+O(\epsilon_{0})}$ .

In place of GRH, one could use an elementary $\ell^{2}$ statement in the spirit of a large sieve inequality. One would then use [B2’] instead of [B1’]. (See [wang2023_large_sieve_diagonal_cubic_forms].) ∎

We now diagnose (and sketch “cures” for) the key “sources of $\epsilon$ ” above:

(1)

In (H3), the pointwise GRH bound $\sum_{N\leq n<2N}\mu_{\bm{c}}(n)\ll_{\epsilon}\lVert\bm{c}\rVert^{\epsilon}N^{1/2+\epsilon}$ . (Cure: RMT-type predictions such as Conjectures 1.4 and 1.8.)
(2)
In (H4), the bound [B1’]. (Or [B2’], for the argument of [wang2023_large_sieve_diagonal_cubic_forms].) In fact, upon closer inspection, the proofs of [B1’] and [B2’] each have two sources of $\epsilon$ :
1. (a)
  
  Good prime factors $p\nmid\Delta(\bm{c})$ of $n$ , via the “first-order error” present in $\Psi^{\bm{c},2}$ . (Cure: Replacing $\Psi^{\bm{c},1}$ with a “better approximation” of $\Phi$ .)
2. (b)
  
  Bad prime factors $p\mid\Delta(\bm{c})$ of $n$ , via the sometimes large failure of square-root cancellation in individual sums of the form $S_{\bm{c}}(p^{l})$ . (Cure: New and old pointwise bounds on $\lvert S_{\bm{c}}(p^{l})\rvert$ , and the average-type Conjecture 1.5.)
(3)
The fact that over $1\ll N\ll Y$ , each dyadic range $N\leq n<2N$ contributes roughly equally to the final bound (2.17); cf. [wang2023_large_sieve_diagonal_cubic_forms]*Remark 5.3. If unaddressed, then the following sources of $\epsilon$ would arise (in our work):
1. (a)
  
  A fatal $\log{X}$ factor in our proof of Theorem 1.3, via summation over $N$ .
2. (b)
  
  A large contribution from relatively small $n$ in our proof of Theorems 1.6 and 1.9, when handling variation of $J_{\bm{c},X}(n)$ over $\bm{c}$ —a step needed when applying Conjecture 1.8 or 1.10 to beat pointwise GRH.
(Cure: New integral bounds that decay, as $n\to 0$ , fairly uniformly over $\bm{c}$ .)

(4)

The lack of “ $\epsilon$ -care” in bounds and “decay cutoffs” for integrals. The best recorded integral estimates (valid at least for some $w$ ) seem to be

(2.19)

J_{\bm{c},X}(n),\,n\cdot\partial{J_{\bm{c},X}(n)}/\partial{n}\ll_{A}(1+\lVert\bm{c}\rVert/X^{1/2})^{-A}\cdot f(2+X\lVert\bm{c}\rVert/n),

with $f(r)=r^{1-m/2}(\log{r})^{m}$ , from [hooley2014octonary]*p. 252, (31). Summing over $n/X^{1-\epsilon}\leq\lVert\bm{c}\rVert\leq X^{1/2}$ carefully, or summing over $X^{1/2}\leq\lVert\bm{c}\rVert\leq X^{1/2+\epsilon}$ carelessly, incurs $\epsilon$ -losses. (Cure: Summing carefully over $\bm{c}$ , and using the “cure to (3)” over small $n$ .)

3. Background on individual $L$ -functions

3.1. Geometric background

We need to give a precise meaning to Conjecture 1.2. We first define the necessary Hasse–Weil $L$ -functions and their local factors, following [serre1969facteurs]. (Another option, not pursued here, would be to follow [taylor2004galois].) This is technical, but allows us to capture “variation in $p$ ” in a representation-theoretic framework. At most primes, the data captured is very concrete; see e.g. (3.4).

For any perfect field $K$ , let $G_{K}\colonequals\operatorname{Gal}(\overline{K}/K)$ . Let $\Gamma_{\mathbb{R}}(s)\colonequals\pi^{-s/2}\Gamma(s/2)$ and $\Gamma_{\mathbb{C}}(s)\colonequals(2\pi)^{-s}\Gamma(s)$ . The case of the Riemann zeta function $\zeta(s)$ in (1.7) is familiar (with $\zeta_{p}(s)=(1-p^{-s})^{-1}$ and $\zeta_{\infty}(s)=\Gamma_{\mathbb{R}}(s)$ ), so we focus on the other cases. Let $\bm{c}\in\mathcal{S}_{1}$ . Let $\ell$ be a prime, and (viewing $V$ , $V_{\bm{c}}$ as subvarieties of $\mathbb{P}^{m-1}$ ) consider the $\ell$ -adic Galois representations

\rho_{V}\colon G_{\mathbb{Q}}\to\frac{H^{1+m_{\ast}}(V\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathbb{Q}_{\ell})}{H^{1+m_{\ast}}(\mathbb{P}^{m-1}_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})},\quad\rho_{V_{\bm{c}}}\colon G_{\mathbb{Q}}\to\frac{H^{m_{\ast}}(V_{\bm{c}}\times_{\mathbb{Q}}\overline{\mathbb{Q}},\mathbb{Q}_{\ell})}{H^{m_{\ast}}(\mathbb{P}^{m-1}_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})}.

It is known that the representations $\rho_{V}$ , $\rho_{V_{\bm{c}}}$ , $\bigotimes^{2}\rho_{V_{\bm{c}}}$ , $\operatorname{Sym}^{2}\rho_{V_{\bm{c}}}$ , $\bigwedge^{2}\rho_{V_{\bm{c}}}$ of $G_{\mathbb{Q}}$ have dimensions depending only on $m$ , and are pure of weight $(1+m_{\ast})/2$ , $m_{\ast}/2$ , $m_{\ast}$ , $m_{\ast}$ , $m_{\ast}$ , respectively.

Let $\rho\colon G_{\mathbb{Q}}\to M$ be one of these five representations, and let $d_{\rho}$ , $w_{\rho}$ be the dimension and weight of $\rho$ , respectively. Define $\Gamma_{\rho}(s)$ using Hodge theory, following [serre1969facteurs]*§3.2 (after passing from $M$ to a singular cohomology group $M^{\prime}/\mathbb{C}$ independent of $\ell$ ); then for certain integers $h_{\rho}(+),h_{\rho}(-),h_{\rho}(a,b)\geq 0$ with $h_{\rho}(+)+h_{\rho}(-)+2\sum_{0\leq a<b:\,a+b=w_{\rho}}h_{\rho}(a,b)=d_{\rho}$ , we have

\Gamma_{\rho}(s)=\Gamma_{\mathbb{R}}(s-w_{\rho}/2)^{h_{\rho}(+)}\Gamma_{\mathbb{R}}(s-w_{\rho}/2+1)^{h_{\rho}(-)}\prod_{0\leq a<b:\,a+b=w_{\rho}}\Gamma_{\mathbb{C}}(s-a)^{h_{\rho}(a,b)}.

Let $L_{\infty}(s,\rho)\colonequals\Gamma_{\rho}(s+w_{\rho}/2)$ ; then $L_{\infty}(s,\rho)$ is holomorphic on the half-plane $\operatorname{Re}(s)>0$ . Note that if we let $\bm{c}$ , $\rho$ vary, the number of possible functions $L_{\infty}(s,\rho)$ could be is $\ll_{m}1$ .

For each prime $p\neq\ell$ , we may restrict $\rho$ to $G_{\mathbb{Q}_{p}}$ ; let $P_{p}(T)\colonequals\det(1-\pi_{p}T)\in\mathbb{Q}_{\ell}[T]$ denote the reverse characteristic polynomial of geometric Frobenius on $M^{I_{\mathbb{Q}_{p}}}$ (the inertia invariants of $M$ ), following [serre1969facteurs]*§2.2. Clearly $\deg P_{p}\leq d_{\rho}$ , with equality if and only if $\rho$ is unramified at $p$ . Write $P_{p}(T)=\prod_{1\leq j\leq\deg P_{p}}(1-\alpha_{\rho,j}(p)T)$ , and let

(3.1)

\tilde{\alpha}_{\rho,j}(p)\colonequals p^{-w_{\rho}/2}\alpha_{\rho,j}(p),\quad L_{p}(s,\rho)\colonequals\prod_{1\leq j\leq\deg P_{p}}(1-\tilde{\alpha}_{\rho,j}(p)p^{-s})^{-1}=P_{p}(p^{-s-w_{\rho}/2})^{-1}.

Because $V$ , $V_{\bm{c}}$ are complete intersections in $\mathbb{P}^{m-1}_{\mathbb{Q}}$ , it is now known⁴⁴4thanks to [laskar2017local]*Corollary 1.2 and its proof (cf. [saito2003weight]*Corollary 0.6), which builds on progress of [scholze2012perfectoid] on the weight-monodromy conjecture that the polynomial $P_{p}(T)$ lies in $\mathbb{Q}[T]$ and is independent of $\ell$ (so that $\{\alpha_{\rho,j}(p)\}_{j}$ is a multiset of algebraic numbers), and furthermore (for any embedding of $\alpha_{\rho,j}(p)$ into the complex numbers) we have

(3.2)

\lvert\alpha_{\rho,j}(p)\rvert\leq p^{w_{\rho}/2},\quad\lvert\tilde{\alpha}_{\rho,j}(p)\rvert\leq 1,

for all $p$ , $j$ . One might also be able to directly (without automorphy) define a conductor and root number for $\rho$ independent of $\ell$ (following [serre1969facteurs] and [deligne1969constantes]), but we need not do so.

Let $L_{\infty}(s,V)\colonequals L_{\infty}(s,\rho_{V})$ for any $\ell$ . Given a prime $p$ , let $L_{p}(s,V)\colonequals L_{p}(s,\rho_{V})$ for any $\ell\neq p$ , and let $\tilde{\alpha}_{V,j}(p)\colonequals\tilde{\alpha}_{\rho_{V},j}(p)$ and $d_{V,p}\colonequals\deg P_{p}$ . Let

L(s,V)\colonequals\prod_{p<\infty}L_{p}(s,V)\equalscolon\sum_{n\geq 1}\lambda^{\natural}_{V}(n)n^{-s},

so that $\lambda^{\natural}_{V}(p)=\sum_{1\leq j\leq d_{V,p}}\tilde{\alpha}_{V,j}(p)$ for all $p$ . Let $\deg L(s,V)\colonequals\max_{p}d_{V,p}=d_{\rho_{V}}$ . Make analogous definitions for $L(s,V_{\bm{c}})$ and its tensor squares in (1.7), in terms of $\rho_{V_{\bm{c}}}$ and its tensor squares. If $p\nmid\Delta(\bm{c})$ and $d=\deg L(s,V_{\bm{c}})$ , then by (2.12), (3.1), smooth proper base change, and the Grothendieck–Lefschetz trace formula, we have (for instance)

(3.3)	$\displaystyle\lambda^{\natural}_{V}(p)$	$\displaystyle=\sum_{1\leq j\leq\deg L(s,V)}\tilde{\alpha}_{V,j}(p)=(-1)^{1+m_{\ast}}E^{\natural}_{F}(p),$
(3.4)	$\displaystyle\lambda^{\natural}_{V_{\bm{c}}}(p)$	$\displaystyle=\sum_{1\leq j\leq d}\tilde{\alpha}_{V_{\bm{c}},j}(p)=(-1)^{m_{\ast}}E^{\natural}_{\bm{c}}(p),$
(3.5)	$\displaystyle\lambda^{\natural}_{V_{\bm{c}},\operatorname{Sym}^{2}}(p)$	$\displaystyle=\sum_{1\leq i\leq j\leq d}\tilde{\alpha}_{V_{\bm{c}},i}(p)\tilde{\alpha}_{V_{\bm{c}},j}(p)=\lambda^{\natural}_{V_{\bm{c}}}(p^{2}),$
(3.6)	$\displaystyle\lambda^{\natural}_{V_{\bm{c}},\bigwedge^{2}}(p)$	$\displaystyle=\sum_{1\leq i<j\leq d}\tilde{\alpha}_{V_{\bm{c}},i}(p)\tilde{\alpha}_{V_{\bm{c}},j}(p)=\lambda^{\natural}_{V_{\bm{c}}}(p)^{2}-\lambda^{\natural}_{V_{\bm{c}}}(p^{2}),$
(3.7)	$\displaystyle(-1)^{m_{\ast}}E^{\natural}_{\bm{c}}(p^{2})$	$\displaystyle=\sum_{1\leq j\leq d}\tilde{\alpha}_{V_{\bm{c}},j}(p)^{2}=E^{\natural}_{\bm{c}}(p)^{2}-2\lambda^{\natural}_{V_{\bm{c}},\bigwedge^{2}}(p),$

where for all $j$ we have (by the Weil conjectures, since $p\nmid\Delta(\bm{c})$ )

(3.8)

\lvert\tilde{\alpha}_{V,j}(p)\rvert=1,\quad\lvert\tilde{\alpha}_{V_{\bm{c}},j}(p)\rvert=1.

Remark 3.1.

By working locally from the beginning, one can define (for any integer $n\geq 1$ ) the quantities $\prod_{p\mid n}L_{p}(s,V_{\bm{c}})$ and $\lambda^{\natural}_{V_{\bm{c}}}(n)$ on all of $\{\bm{c}\in\prod_{p\mid n}\mathbb{Z}_{p}^{m}:\Delta(\bm{c})\neq 0\}$ . Extended definitions like this will be convenient for local calculations in §6.1 and §7.3.

3.2. Automorphic background

We need some background on automorphic representations $\Pi$ of $\operatorname{GL}_{d}(\mathbf{A}_{\mathbb{Q}})$ for $d\geq 1$ . We will only work with cuspidal $\Pi$ ’s, or more generally, isobaric $\Pi$ ’s. These $\Pi$ ’s have well-defined $L$ -functions $L(s,\Pi)$ , and good formal properties (due to Rankin, Selberg, Langlands, Godement, Jacquet, Shalika, and others):

(1)

If $\Pi$ is cuspidal, then $L(s,\Pi)$ is primitive in the sense of [farmer2019analytic]*Lemma 2.4, and has certain familiar analytic properties [farmer2019analytic]*Theorem 3.1.
(2)

For each isobaric $\Pi$ , there is a unique multiset $\{\Pi_{1},\dots,\Pi_{r}\}$ , consisting of cuspidals, such that $L(s,\Pi)=L(s,\Pi_{1})\cdots L(s,\Pi_{r})$ . We call the $\Pi_{i}$ ’s cuspidal constituents of $\Pi$ .
(3)

Strong multiplicity one: If $\Pi$ , $\Pi^{\prime}$ are isobaric, and $L_{p}(s,\Pi)=L_{p}(s,\Pi^{\prime})$ for all but finitely many primes $p$ , then $L(s,\Pi)=L(s,\Pi^{\prime})$ .

Conjecture 1.2 has a host of standard consequences (which may be treated as a black box).

Proposition 3.2.

Let $\bm{c}\in\mathcal{S}_{1}$ . Let $L(s)$ be one of the Hasse–Weil $L$ -functions in (1.7). Assume Conjecture 1.2 for $L(s)$ holds for some $d$ , $\Pi$ . Then the following hold:

(1)

$d=\deg L(s)$ ; in particular, $d\ll_{m}1$ .
(2)

The conductor $q(\Pi)\in\mathbb{Z}_{\geq 1}$ of $\Pi$ satisfies $q(\Pi)\mid\Delta(\bm{c})^{O_{m}(1)}$ .
(3)

Each cuspidal constituent of $\Pi$ has unitary, finite-order central character.
(4)

$L(s)$ is holomorphic on $\mathbb{C}$ , except possibly for poles at $s=1$ corresponding to trivial constituents of $\Pi$ .
(5)

$(s-1)^{d}L(s)$ is an entire function of order $1$ .
(6)

$L(s)$ has a standard functional equation (with critical line $\operatorname{Re}(s)=1/2$ ), involving $q(\Pi)$ and some root number $\varepsilon(\Pi)\in\{z\in\mathbb{C}:\lvert z\rvert=1\}$ .
(7)

$L(s)$ has real coefficients, $\Pi$ is self-dual, and $\varepsilon(\Pi)\in\{-1,+1\}$ .
(8)

Let $\psi$ be a cuspidal or isobaric constituent of $\Pi$ . Then $1/L(s,\psi)\ll_{m,\epsilon}\lVert\bm{c}\rVert^{\epsilon}(1+\lvert s\rvert)^{\epsilon}$ for $\operatorname{Re}(s)\geq 1/2+\epsilon$ . If $\mu_{\psi}(n)$ denotes the $n$ th coefficient of the Dirichlet series $1/L(s,\psi)$ , then $\sum_{1\leq n\leq N}\mu_{\psi}(n)n^{-it}\ll_{m,\epsilon}\lVert\bm{c}\rVert^{\epsilon}(1+\lvert t\rvert)^{\epsilon}N^{1/2+\epsilon}$ for all $t\in\mathbb{R}$ and $N\in\mathbb{R}_{>0}$ .

Proof.

If $p$ is a prime, then $L_{p}(s,\Pi)$ has degree $\leq d$ , with equality if and only if $p\nmid q(\Pi)$ ; cf. [farmer2019analytic]*Axiom 3(b) and (3.3).

(1): Compare the degrees of $L_{p}(s)$ , $L_{p}(s,\Pi)$ at a prime $p\nmid q(\Pi)\Delta(\bm{c})$ .

(2): For some $\nu\in\mathbb{Z}$ , the local Dirichlet polynomial $L_{p}(s-\nu/2)$ has rational coefficients for all primes $p$ . So by [shin2014fields]*(3.2), there exists $u\in\mathbb{R}$ such that for all $p\nmid q(\Pi)$ , the representation $\Pi_{p}\otimes(\lvert\cdot\rvert^{u}\circ\det)$ of $\operatorname{GL}_{d}(\mathbb{Q}_{p})$ has field of rationality $\mathbb{Q}$ , in the sense of [shin2014fields]*Definition 2.2. So by strong multiplicity one, the field of rationality of $\Pi\otimes(\lvert\cdot\rvert^{u}\circ\det)$ is $\mathbb{Q}$ . Now consider any $p\mid q(\Pi)$ . Then $L_{p}(s,\Pi)$ has degree $<d$ , so $L_{p}(s)$ has degree $<d$ , whence $p\mid\Delta(\bm{c})$ . By [shin2014fields]*Lemmas 3.11 and 3.13, then, $v_{p}(q(\Pi))\ll_{d}1$ . So $q(\Pi)\mid\Delta(\bm{c})^{O_{d}(1)}\mid\Delta(\bm{c})^{O_{m}(1)}$ .

(3): Let $\Pi^{\prime}$ be a cuspidal constituent of $\Pi$ , so $\Pi^{\prime}$ is a cuspidal automorphic representation of $\operatorname{GL}_{d^{\prime}}(\mathbf{A}_{\mathbb{Q}})$ for some $d^{\prime}\geq 1$ . Let $\omega^{\prime}\colon\mathbb{Q}^{\times}\backslash\mathbf{A}_{\mathbb{Q}}^{\times}\to\mathbb{C}^{\times}$ be the central character of $\Pi^{\prime}$ . Then $\omega^{\prime}$ corresponds to a classical character $n\mapsto\lvert n\rvert^{z}\chi(n)$ on $\mathbb{Z}$ , where $z\in\mathbb{C}$ and $\chi$ is a Dirichlet character of conductor dividing $q(\Pi)$ , such that $\omega^{\prime}_{p}(p)=p^{z}\chi(p)$ for all primes $p\nmid q(\Pi)$ . But at each prime $p\nmid q(\Pi)$ , if we write $L_{p}(s,\Pi)=\prod_{1\leq j\leq d^{\prime}}(1-\alpha_{\Pi,j}(p)p^{-s})^{-1}$ , then $\omega^{\prime}_{p}(p)=\prod_{1\leq j\leq d^{\prime}}\alpha_{\Pi,j}(p)$ ; cf. [farmer2019analytic]*(3.3) and its proof. By (3.8) and the algebraicity of the eigenvalues $\tilde{\alpha}_{j}(p)$ , it follows that for infinitely many primes $p$ , we have $\lvert p^{z}\rvert=1$ and $p^{z}\in\overline{\mathbb{Q}}$ . So $\operatorname{Re}(z)=0$ , i.e. $\omega^{\prime}$ is unitary; and then $\operatorname{Im}(z)=0$ by the six exponentials theorem (cf. [farmer2019analytic]*proof of Lemma 4.9), so $\omega^{\prime}$ has finite order.

(4), (5), (6): Use (3) and results of Godement and Jacquet; cf. [farmer2019analytic]*Theorem 3.1.

(7): For some $\nu\in\mathbb{Z}$ , the coefficients of $L(s-\nu/2)$ are all rational. Hence $L(s)$ has real coefficients. So by (3) and strong multiplicity one, we have $L(s,\Pi)=L(s,\Pi^{\vee})$ and thus $\Pi$ is self-dual. The functional equation from (6) then implies $\varepsilon(\Pi)\in\mathbb{R}$ , so $\varepsilon(\Pi)=\pm 1$ .

(8): By (3.2), (5), (6), and Conjecture 1.2(2), we have $1/L(s,\psi)\ll_{d,\epsilon}q(\psi)^{\epsilon}(1+\lvert s\rvert)^{\epsilon}$ for $\operatorname{Re}(s)\geq 1/2+\epsilon$ ; see e.g. [iwaniec2004analytic]*Theorem 5.19 and the ensuing paragraph. But $q(\psi)\mid q(\Pi)\mid\Delta(\bm{c})^{O_{m}(1)}$ , so the desired bound on $1/L(s,\psi)$ follows. One can then prove $\sum_{1\leq n\leq N}\mu_{\psi}(n)n^{-it}\ll_{m,\epsilon}\lVert\bm{c}\rVert^{\epsilon}(1+\lvert t\rvert)^{\epsilon}N^{1/2+\epsilon}$ using Perron’s formula ([iwaniec2004analytic]*Proposition 5.54) and contour integration; cf. [hooley1986HasseWeil]*p. 75, proof of Lemma 10. ∎

4. Local control on polynomials and $L$ -functions

We first recall some standard bounds on the local near-zero loci of a fixed polynomial; cf. [serre1981quelques]*p. 146, (57) and [ganzburg2001polynomial]. Given $f\in\mathbb{Z}[x]$ , let $N(f;q)$ be the number of solutions $x\bmod{q}$ to $f(x)\equiv 0\bmod{q}$ , and let $\mu_{\mathbb{R}}(f;\lambda)$ be the Lebesgue measure of the set $\{x\in[-1,1]:\lvert f(x)\rvert\leq\lambda\}$ . Similarly, for $P\in\mathbb{Z}[y_{1},\dots,y_{n}]$ , define

\begin{split}N(P;q)&\colonequals\#{\{(y_{1},\dots,y_{n})\in(\mathbb{Z}/q\mathbb{Z})^{n}:P(y_{1},\dots,y_{n})\equiv 0\bmod{q}\}},\\ \mu_{\mathbb{R}}(P;\lambda)&\colonequals\operatorname{vol}{\{(y_{1},\dots,y_{n})\in[-1,1]^{n}:\lvert P(y_{1},\dots,y_{n})\rvert\leq\lambda\}}.\end{split}

Proposition 4.1.

Suppose $f\in\mathbb{Z}[x]$ has leading term $ax^{d}$ with $a\neq 0$ and $d\geq 1$ . Then

(4.1)

N(f;q)\ll_{d}\lvert a\rvert^{1/d}q^{1-1/d},

uniformly over integers $q\geq 1$ . Also, uniformly over reals $\lambda>0$ , we have

(4.2)

\mu_{\mathbb{R}}(f;\lambda)\ll_{d}\lvert a\rvert^{-1/d}\lambda^{1/d}.

Proof.

The bound (4.2) goes back to Pólya (see e.g. [ganzburg2001polynomial]*Theorem 1.1). Now let $h$ denote the greatest common divisor of $q$ and the coefficients of $f$ . The bound (4.1) follows from [konjagin1979number] if $h=1$ , and then in general from the inequality $N(f;q)\leq h\cdot N(f/h;q/h)$ . ∎

Corollary 4.2.

Fix a nonconstant polynomial $P\in\mathbb{Z}[y_{1},\dots,y_{n}]$ , where $n\geq 1$ . Then

(4.3)

N(P;q)\ll_{P}q^{n-1/\deg{P}},

uniformly over integers $q\geq 1$ . Also, uniformly over reals $\lambda>0$ , we have

(4.4)

\mu_{\mathbb{R}}(P;\lambda)\ll_{P}\lambda^{1/\deg{P}}.

Proof.

Let $d=\deg{P}$ . Given a matrix $A\in\operatorname{GL}_{n}(\mathbb{Q})$ with integral entries, let $Q=(\det{A})^{d}\cdot P\circ A^{-1}\in\mathbb{Z}[y_{1},\dots,y_{n}]$ . Via the $\mathbb{Z}$ -linear map $\bm{y}\mapsto A\bm{y}$ , we have $N(P;q)\ll_{A}N(Q;q)$ and $\mu_{\mathbb{R}}(P;\lambda)\ll_{A}\mu_{\mathbb{R}}(Q;\lambda)$ . By [eisenbud1995commutative]*p. 283, proof of Lemma 13.2.c, we may choose $A$ so that $Q$ has $y_{1}$ -leading term $a_{1}y_{1}^{d}$ , with $a_{1}\neq 0$ . Fix $y_{2},\dots,y_{n}$ ; then $\#\{y_{1}:Q(\bm{y})\equiv 0\}\ll_{d,a_{1}}q^{1-1/d}$ by (4.1). Summing over $y_{2},\dots,y_{n}$ gives (4.3). Similarly, (4.2) implies (4.4). ∎

We now turn to local $L$ -factors. (For $\bm{c}\in\mathbb{Z}_{p}^{m}$ with $\Delta(\bm{c})\neq 0$ , we define $L_{p}(s,V_{\bm{c}})$ using Remark 3.1.)

Proposition 4.3 ([kisin1999local]).

Fix a prime $p$ and a tuple $\bm{b}\in\mathbb{Z}_{p}^{m}$ with $\Delta(\bm{b})\neq 0$ . Then there exists an integer $l\geq 0$ , depending only on $p$ and $\bm{b}$ , such that for all tuples $\bm{a}\in\mathbb{Z}_{p}^{m}$ with $\bm{a}\equiv\bm{b}\bmod{p^{1+l}}$ , we have $\Delta(\bm{a})\neq 0$ and $L_{p}(s,V_{\bm{a}})=L_{p}(s,V_{\bm{b}})$ .

Proof.

Let $S$ be the open subscheme $\Delta(\bm{c})\neq 0$ of $\mathbb{A}^{m}_{\mathbb{Q}_{p}}$ . Let $W$ be the closed subscheme $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ of $\mathbb{P}^{m-1}_{S}$ . Let $\ell\neq p$ be a prime. The maps $f_{1}\colon W\to S$ and $f_{2}\colon\mathbb{P}^{m-1}_{S}\to S$ induce local systems $\mathcal{L}_{j}\colonequals R^{m_{\ast}}(f_{j})_{\ast}(\mathbb{Z}_{\ell})$ on $S$ . By [kisin1999local]*Theorem 5.1, case (2), and its proof, there exists a $p$ -adic neighborhood $U$ of $\bm{b}$ in $S$ such that the Galois representations $\rho_{j,\bm{a}}\colon G_{\mathbb{Q}_{p}}\to\operatorname{Aut}(\mathcal{L}_{j,\bm{a}})$ for $\bm{a}\in U(\mathbb{Q}_{p})$ all factor through $\pi_{1}(U)$ in an appropriate sense. So the isomorphism class of the representation $G_{\mathbb{Q}_{p}}\to\operatorname{Aut}(\mathbb{Q}_{\ell}\otimes(\mathcal{L}_{1,\bm{a}}/\mathcal{L}_{2,\bm{a}}))$ is constant over $\bm{a}\in U(\mathbb{Q}_{p})$ , and thus $L_{p}(s,V_{\bm{a}})$ is too. ∎

Lemma 4.4.

Fix a prime $p$ . For each $\bm{b}\in\mathbb{Z}_{p}^{m}$ with $\Delta(\bm{b})\neq 0$ , let $l(p,\bm{b})\geq 0$ be the smallest integer for which the conclusion of Proposition 4.3 holds. Then for each integer $l\geq 0$ , the set

(4.5)

\{\bm{b}\in\mathbb{Z}_{p}^{m}:\Delta(\bm{b})\neq 0,\;l(p,\bm{b})\leq l\}

is invariant under translation by any element of $p^{l+1}\mathbb{Z}_{p}^{m}$ . Furthermore, the measure of (4.5) tends to $1$ as $l\to\infty$ . (Here we use the usual Haar measure on $\mathbb{Z}_{p}^{m}$ .)

Proof.

Suppose $\bm{b},\bm{c}\in\mathbb{Z}_{p}^{m}$ with $\Delta(\bm{b})\neq 0$ and $\bm{c}\equiv\bm{b}\bmod{p^{1+l(p,\bm{b})}}$ . Then $\Delta(\bm{c})\neq 0$ and $L_{p}(s,V_{\bm{c}})=L_{p}(s,V_{\bm{b}})$ , and thus $l(p,\bm{c})\leq l(p,\bm{b})$ (because for every $\bm{a}\equiv\bm{c}\bmod{p^{1+l(p,\bm{b})}}$ , we have $\bm{a}\equiv\bm{b}\bmod{p^{1+l(p,\bm{b})}}$ and thus $\Delta(\bm{a})\neq 0$ and $L_{p}(s,V_{\bm{a}})=L_{p}(s,V_{\bm{b}})=L_{p}(s,V_{\bm{c}})$ ). But then $\bm{b}\equiv\bm{c}\bmod{p^{1+l(p,\bm{c})}}$ , so a similar argument gives $l(p,\bm{b})\leq l(p,\bm{c})$ , whence

(4.6)

l(p,\bm{b})=l(p,\bm{c}).

Therefore, if $\bm{b}$ lies in (4.5) for some $l\geq 0$ , then (4.5) indeed contains the set $\bm{b}+p^{l+1}\mathbb{Z}_{p}^{m}$ .

Next, let $A\in\mathbb{Z}_{\geq 0}$ . The set $S_{A}=\{\bm{c}\in\mathbb{Z}_{p}^{m}:v_{p}(\Delta(\bm{c}))\leq A\}$ is closed in $\mathbb{Z}_{p}^{m}$ , and thus compact. The function $\bm{b}\mapsto l(p,\bm{b})$ on $S_{A}$ is locally constant (by (4.6)), and thus has a (finite) maximum value. Therefore, for all $l$ sufficiently large in terms of $A$ , the set (4.5) contains $S_{A}$ . But by (4.3) (or by [serre1981quelques]*p. 146, Corollaire), the measure of $S_{A}$ tends to $1$ as $A\to\infty$ . ∎

Remark 4.5.

If Conjecture 1.11 holds, then $l(p,\bm{b})\leq v_{p}(H(\bm{b}))$ (whenever $H(\bm{b})\neq 0$ ).

5. General separation technique

At several points in the paper, we need to understand quantities that vary with $\bm{c}$ and $n$ . A key tool we use for this is smooth dyadic decomposition (minimizing convergence issues) followed by separation of variables (via Mellin inversion); see Lemma 5.2.

Let $d^{\times}{r}\colonequals dr/r$ and $\partial_{\log{r}}\colonequals r\cdot\partial_{r}$ for $r\in\mathbb{R}_{>0}$ . For any $k\in\mathbb{Z}_{\geq 1}$ and $\bm{r}\in\mathbb{R}_{>0}^{k}$ , let $d^{\times}{\bm{r}}=d^{\times}{r_{1}}\cdots d^{\times}{r_{k}}$ and $\partial_{\log{\bm{r}}}^{\bm{\alpha}}\colonequals\partial_{\log{r_{1}}}^{\alpha_{1}}\cdots\partial_{\log{r_{k}}}^{\alpha_{k}}$ . Given $g\in C^{\infty}_{c}(\mathbb{R}_{>0}^{k})\otimes\mathbb{C}$ and $\bm{s}\in\mathbb{C}^{k}$ , let

(5.1)

g^{\vee}(\bm{s})\colonequals\int_{\bm{r}>0}d^{\times}\bm{r}\,g(\bm{r})\bm{r}^{\bm{s}}=\int_{r_{1},\dots,r_{k}>0}d^{\times}{r_{1}}\cdots d^{\times}{r_{k}}\,g(r_{1},\dots,r_{k})r_{1}^{s_{1}}\cdots r_{k}^{s_{k}},

so that Mellin inversion (see e.g. [iwaniec2004analytic]*p. 90, (4.106)) gives (for all $\bm{\sigma}\in\mathbb{R}^{k}$ )

(5.2)

g(\bm{r})=(2\pi)^{-k}\int_{\bm{t}\in\mathbb{R}^{k}}d\bm{t}\,g^{\vee}(\bm{\sigma}+i\bm{t})\cdot\bm{r}^{-\bm{\sigma}-i\bm{t}}.

Proposition 5.1 (Standard Mellin bound).

Fix a compact set $I\subseteq\mathbb{R}_{>0}$ . Let $k\in\mathbb{Z}_{\geq 1}$ and $(\bm{M},\bm{s})\in\mathbb{R}_{>0}^{k}\times\mathbb{C}^{k}$ . Let $g\in C^{\infty}_{c}(\mathbb{R}_{>0}^{k})\otimes\mathbb{C}$ with $\operatorname{Supp}{g}\subseteq\prod_{1\leq i\leq k}(M_{i}\cdot I)$ . Then

(5.3)

g^{\vee}(\bm{s})\ll_{k,b}O_{I}(1)^{k}\cdot\frac{\lVert g\circ\exp\rVert_{b,\infty}}{(1+\lVert\bm{s}\rVert)^{b}}\cdot\prod_{1\leq i\leq k}(O_{I}(1)^{\lvert\operatorname{Re}(s_{i})\rvert}M_{i}^{\operatorname{Re}(s_{i})}),

for all $b\in\mathbb{Z}_{\geq 0}$ . Here $\lVert g\circ\exp\rVert_{b,\infty}\colonequals\sum_{\lvert\bm{\alpha}\rvert\leq b}\sup_{\bm{r}\in\mathbb{R}_{>0}^{k}}{\lvert\partial_{\log\bm{r}}^{\bm{\alpha}}g(\bm{r})\rvert}$ .

Proof.

By (5.1), and the scale invariance of $d^{\times}{r_{i}}$ , $\partial_{\log{r_{i}}}$ for each $i$ , we can reduce to the case where $M_{1}=\cdots=M_{k}=1$ . Now let $\operatorname{vol}(\log{I})\colonequals\int_{r>0}d^{\times}{r}\,\bm{1}_{r\in I}<\infty$ . If $\lVert\bm{s}\rVert\leq 1$ , we may assume $b=0$ . Now in general, choose $i$ with $\lvert s_{i}\rvert=\max(\lvert s_{1}\rvert,\dots,\lvert s_{d}\rvert)$ . Then on the right-hand side of (5.1), integrate by parts $b$ times in $\log{r_{i}}$ , to rewrite $g^{\vee}(\bm{s})$ as

(-1)^{b}\int_{\bm{r}>0}d^{\times}{\bm{r}}\,\frac{\bm{r}^{\bm{s}}}{s_{i}^{b}}\partial_{\log{r_{i}}}^{b}g(\bm{r})\ll\operatorname{vol}(\log{I})^{k}\frac{\lVert g\circ\exp\rVert_{b,\infty}}{\lvert s_{i}\rvert^{b}}\prod_{1\leq j\leq k}O_{\operatorname{Supp}{I}}(1)^{\lvert\operatorname{Re}(s_{j})\rvert}.

This suffices for (5.3). ∎

Fix a function $\nu_{2}\in C^{\infty}_{c}(\mathbb{R}_{>0})$ , supported on $[1,2]$ , with

(5.4)

\int_{r>0}d^{\times}{r}\,\nu_{2}(r)^{2}=1.

Lemma 5.2 (“Dyadic partial Mellin summation”).

Let $k\in\mathbb{Z}_{\geq 1}$ . Let $a\colon\mathbb{Z}_{\geq 1}^{k}\to\mathbb{C}$ be a function. Let $f\colon\mathbb{R}_{>0}^{k}\to\mathbb{C}$ be a smooth function supported on $\prod_{1\leq j\leq k}r_{j}\leq A$ for some real $A\geq 1$ . Let $\nu=\nu_{2}$ ; let $\nu(\bm{r}/\bm{N})\colonequals\prod_{1\leq j\leq k}\nu(r_{j}/N_{j})$ and $g_{\bm{N}}(\bm{r})\colonequals f(\bm{r})\nu(\bm{r}/\bm{N})$ . Then

(5.5)

\sum_{\bm{n}\geq 1}a(\bm{n})f(\bm{n})=(2\pi)^{-k}\int_{\bm{N}\in[1/2,\infty)^{k}}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{k}}d\bm{t}\,g_{\bm{N}}^{\vee}(i\bm{t})\sum_{\bm{n}\geq 1}\nu(\bm{n}/\bm{N})a(\bm{n})\bm{n}^{i\bm{t}}.

Proof.

For all $\bm{n}\geq 1$ , we have $f(\bm{n})=\int_{\bm{N}\geq 1/2}d^{\times}{\bm{N}}\,\nu(\bm{n}/\bm{N})g_{\bm{N}}(\bm{n})$ by (5.4) (since $\nu(\bm{n}/\bm{N})=0$ for $\bm{N}<1/2$ ). Also, $g_{\bm{N}}(\bm{r})\in C^{\infty}_{c}(\mathbb{R}_{>0}^{k})\otimes\mathbb{C}$ for each $\bm{N}\geq 1/2$ . So by (5.2), we get

(5.6)

\sum_{\bm{n}\geq 1}a(\bm{n})f(\bm{n})=\sum_{\bm{n}\geq 1}a(\bm{n})\int_{\bm{N}\geq 1/2}d^{\times}\bm{N}\,\nu(\bm{n}/\bm{N})(2\pi)^{-k}\int_{\bm{t}\in\mathbb{R}^{k}}d\bm{t}\,g_{\bm{N}}^{\vee}(i\bm{t})\bm{n}^{i\bm{t}}.

If $\bm{r}\in\operatorname{Supp}{g_{\bm{N}}}$ , then $\prod_{j}r_{j}\leq A$ and $\bm{r}\in\prod_{j}[N_{j},2N_{j}]$ , so $\prod_{j}N_{j}\leq A$ . Therefore, there exist compact sets $\mathcal{K}_{1},\mathcal{K}_{2}\subseteq\mathbb{R}_{>0}^{k}$ such that if $(\bm{n},\bm{N})\notin\mathcal{K}_{1}\times\mathcal{K}_{2}$ in (5.6), then $\nu(\bm{n}/\bm{N})\cdot g_{\bm{N}}^{\vee}(i\bm{t})=0$ . Furthermore, there exists a compact set $\mathcal{K}_{3}\subseteq\mathbb{R}_{>0}^{k}$ such that if $\bm{N}\in\mathcal{K}_{2}$ , then $\operatorname{Supp}{g_{\bm{N}}}\subseteq\mathcal{K}_{3}$ . So Proposition 5.1 gives $g_{\bm{N}}^{\vee}(i\bm{t})\ll_{f,\nu,b}(1+\lVert\bm{t}\rVert)^{-b}$ for all $b\geq 0$ , uniformly over $\bm{N}\in\mathcal{K}_{2}$ . Thus $\sum_{\bm{n}\geq 1}\int_{\bm{N}\geq 1/2}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{k}}d\bm{t}\,\lvert a(\bm{n})\rvert\cdot\lvert\nu(\bm{n}/\bm{N})\rvert\cdot\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert$ is $\ll_{a,f,\nu,b}\sum_{\bm{n}\in\mathcal{K}_{1}\cap\mathbb{Z}}\int_{\bm{N}\in\mathcal{K}_{2}}d^{\times}\bm{N}<\infty$ , provided $b>k$ . Now (5.5) follows from (5.6) by Fubini. ∎

For later use, we now do a general dyadic calculation.

Lemma 5.3.

Let $q,a,b\in\mathbb{R}$ with $a\leq q<b$ . Let $r_{1},r_{2},\tau\in\mathbb{R}_{>0}$ . Then

\sum_{r_{1}\leq r\leq r_{2}:\,\log_{2}(r)\in\mathbb{Z}}\frac{r^{q}}{r^{a}+\tau^{b-a}r^{b}},\quad\int_{r_{1}\leq r\leq r_{2}}d^{\times}{r}\,\frac{r^{q}}{r^{a}+\tau^{b-a}r^{b}},

are both $\ll_{q,a,b}\min(\tau^{-1},r_{2})^{q-a}\cdot(\log(1+r_{2}/r_{1}))^{\bm{1}_{q=a}}$ .

Proof.

We may assume $r_{1}\leq r_{2}$ , or else the sum and integral both vanish. By subtracting $q$ , $a$ , $b$ by $a$ , we may also assume $a=0$ . If $r_{2}\leq\tau^{-1}$ , then the result follows from the bound $1+\tau^{b}r^{b}\geq 1$ and a geometric series (or corresponding integral). If $r_{2}>\tau^{-1}$ , then separately considering $r\leq\tau^{-1}$ and $r>\tau^{-1}$ leads to the result. ∎

6. Statistics of families of $L$ -functions

Throughout §6, assume $m$ is even. Recall the eigenvalue and coefficient notation from §3.

6.1. Computing local averages

For convenience, let $\tilde{\alpha}_{\bm{c},j}(p)\colonequals\tilde{\alpha}_{V_{\bm{c}},j}(p)$ and $\lambda^{\natural}_{\bm{c}}(n)\colonequals\lambda^{\natural}_{V_{\bm{c}}}(n)$ . Let $\mu_{\bm{c}}(n)$ be the $n$ th coefficient of the Dirichlet series $1/L(s,V_{\bm{c}})$ . We have

(6.1)

1/L_{p}(s,V_{\bm{c}})={\textstyle\prod_{j}(1-\tilde{\alpha}_{\bm{c},j}(p)p^{-s})}

by (3.1). So if $p\nmid\Delta(\bm{c})$ , then (3.4), (3.6), (3.7), and $2\nmid m_{\ast}$ imply

(6.2)

\mu_{\bm{c}}(p)=-\lambda^{\natural}_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p),\quad\mu_{\bm{c}}(p^{2})=\lambda^{\natural}_{V_{\bm{c}},\bigwedge^{2}}(p)=\tfrac{1}{2}(E^{\natural}_{\bm{c}}(p)^{2}+E^{\natural}_{\bm{c}}(p^{2})).

The local statistics of $\lambda^{\natural}_{\bm{c}}(n)$ , $\mu_{\bm{c}}(n)$ over $\bm{c}$ play a basic role in the global statistics of $L(s,V_{\bm{c}})$ over $\bm{c}$ . To prove that certain averages exist, we will use (3.2) and Lemma 4.4. But to estimate said averages, we will take a point-counting approach (though one could use monodromy groups instead; see e.g. [sarnak2016families]*§2.11). The result is Proposition 6.1 below.

Let $\mathcal{B}\subseteq\mathbb{R}^{m}$ be a region of the form $I_{1}\times\cdots\times I_{m}$ , where $I_{1},\dots,I_{m}\subseteq\mathbb{R}$ are compact intervals of positive length. Let $\bm{a}\in\mathbb{Z}^{m}$ and $n_{0},n,n_{1},n_{2}\in\mathbb{Z}_{\geq 1}$ . Let $\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in S}[f]$ be the average of $f$ over $\{\bm{c}\in S:\bm{c}\equiv\bm{a}\bmod{n_{0}}\}$ (assuming this set is nonempty). Let $\mathbb{E}^{\bm{a},n_{0}}_{1\leq\bm{c}\leq n}[f]\colonequals\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\{1,2,\dots,n\}^{m}}[f]$ .

Proposition 6.1 (LocAv).

The following two limits exist, and are independent of $\mathcal{B}$ :

\begin{split}\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n)&\colonequals\lim_{Z\to\infty}\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}}[\mu_{\bm{c}}(n)],\\ \bar{\mu}_{F,2}^{\bm{a},n_{0}}(n_{1},n_{2})&\colonequals\lim_{Z\to\infty}\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}}[\mu_{\bm{c}}(n_{1})\mu_{\bm{c}}(n_{2})].\end{split}

The quantity $\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n)$ is multiplicative in $n$ : if $\gcd(n,n^{\prime})=1$ , then

(6.3)

\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n)\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n^{\prime})=\bar{\mu}_{F,1}^{\bm{a},n_{0}}(nn^{\prime}).

The quantity $\bar{\mu}_{F,2}^{\bm{a},n_{0}}(n_{1},n_{2})$ is multiplicative in $(n_{1},n_{2})$ : if $\gcd(n_{1}n_{2},n^{\prime}_{1}n^{\prime}_{2})=1$ , then

(6.4)

\bar{\mu}_{F,2}^{\bm{a},n_{0}}(n_{1},n_{2})\bar{\mu}_{F,2}^{\bm{a},n_{0}}(n^{\prime}_{1},n^{\prime}_{2})=\bar{\mu}_{F,2}^{\bm{a},n_{0}}(n_{1}n^{\prime}_{1},n_{2}n^{\prime}_{2}).

Now let $p$ be a prime, and let $l,l_{1},l_{2}\geq 0$ be integers. Then (uniformly over $p,l,l_{1},l_{2},\bm{a},n_{0}$ )

(6.5)

\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{l})\ll_{\epsilon}p^{l\epsilon},\quad\bar{\mu}_{F,2}^{\bm{a},n_{0}}(p^{l_{1}},p^{l_{2}})\ll_{\epsilon}p^{(l_{1}+l_{2})\epsilon}.

Furthermore, if $p\nmid n_{0}$ , then

(6.6)		$\displaystyle\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p)=\lambda^{\natural}_{V}(p)p^{-1/2}+O(p^{-1}),\quad\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{2})=1+O(p^{-1}),$
(6.7)		$\displaystyle\bar{\mu}_{F,2}^{\bm{a},n_{0}}(p^{l},1)=\bar{\mu}_{F,2}^{\bm{a},n_{0}}(1,p^{l})=\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{l}),\quad\bar{\mu}_{F,2}^{\bm{a},n_{0}}(p,p)=1+O(p^{-1}).$

Proof.

For convenience, define $\mu_{\bm{c}}(n)\bm{1}_{\Delta(\bm{c})\neq 0}$ to be $\mu_{\bm{c}}(n)$ if $\Delta(\bm{c})\neq 0$ , and $0$ if $\Delta(\bm{c})=0$ . (Here we allow $\bm{c}\in\mathbb{Z}^{m}$ , or more generally, $\bm{c}\in\prod_{p\mid n}\mathbb{Z}_{p}^{m}$ .) By (3.2) and (6.1), we have

(6.8)

\mu_{\bm{c}}(p^{l})\bm{1}_{\Delta(\bm{c})\neq 0}\ll_{m}1,\quad\mu_{\bm{c}}(n)\bm{1}_{\Delta(\bm{c})\neq 0}\ll_{\epsilon}n^{\epsilon}.

(In contrast, for $\lambda^{\natural}_{\bm{c}}$ , we have $\lambda^{\natural}_{\bm{c}}(p^{l})\bm{1}_{\Delta(\bm{c})\neq 0}\ll(l+1)^{O_{m}(1)}\ll_{m,\epsilon}p^{l\epsilon}$ and $\lambda^{\natural}_{\bm{c}}(n)\bm{1}_{\Delta(\bm{c})\neq 0}\ll_{\epsilon}n^{\epsilon}$ .)

Since $\mu_{\bm{c}}(1)\bm{1}_{\Delta(\bm{c})\neq 0}=\bm{1}_{\Delta(\bm{c})\neq 0}$ , we have $\bar{\mu}_{F,1}^{\bm{a},n_{0}}(1)=\bar{\mu}_{F,2}^{\bm{a},n_{0}}(1,1)=1$ (since $\lvert\mathcal{S}_{0}\cap Z\cdot\mathcal{B}\rvert=o_{\mathcal{B};Z\to\infty}(Z^{m})$ by (2.4)). On the other hand, if $n\geq 2$ and $k\geq 1$ , then by Lemma 4.4, the quantity $\mu_{\bm{c}}(n)\bm{1}_{\Delta(\bm{c})\neq 0}$ depends only on $\bm{c}\bmod{n^{k}}$ , unless $\bm{c}$ lies in one of $o_{n;k\to\infty}(n^{km})$ exceptional residue classes of $\mathbb{Z}^{m}$ (or $\prod_{p\mid n}\mathbb{Z}_{p}^{m}$ ) modulo $n^{k}$ . This, together with (6.8) and the Chinese remainder theorem, implies that (for any $n\geq 1$ ) the three quantities

\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n),\quad\lim_{k\to\infty}\mathbb{E}^{\bm{a},n_{0}}_{1\leq\bm{c}\leq n^{k}}[\mu_{\bm{c}}(n)\bm{1}_{\Delta(\bm{c})\neq 0}],\quad\prod_{p\mid n}\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\mu_{\bm{c}}(p^{v_{p}(n)})\bm{1}_{\Delta(\bm{c})\neq 0}]

all exist and equal one another. Similarly, the following exist and equal one another:

\bar{\mu}_{F,2}^{\bm{a},n_{0}}(n_{1},n_{2}),\quad\lim_{k\to\infty}\mathbb{E}^{\bm{a},n_{0}}_{1\leq\bm{c}\leq(n_{1}n_{2})^{k}}[\mu_{\bm{c}}(n_{1})\mu_{\bm{c}}(n_{2})\bm{1}_{\Delta(\bm{c})\neq 0}],\quad\prod_{p\mid n_{1}n_{2}}\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\mu_{\bm{c}}(p^{v_{p}(n_{1})})\mu_{\bm{c}}(p^{v_{p}(n_{2})})\bm{1}_{\Delta(\bm{c})\neq 0}].

This establishes the required existence, independence, and multiplicativity of limits.

We now turn to the required estimates. First, (6.5) follows from (6.8). Now assume $p\nmid n_{0}$ ; we must prove (6.6) and (6.7). But $p\nmid n_{0}$ implies $\{\bm{c}\in\mathbb{Z}_{p}^{m}:\bm{c}\equiv\bm{a}\bmod{n_{0}}\}=\mathbb{Z}_{p}^{m}$ , so $\mathbb{E}^{\bm{a},n_{0}}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[f]=\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[f]$ for any quantity $f$ . So by our $p$ -adic interpretations (from the previous paragraph) of $\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{l})$ , $\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{l_{1}},p^{l_{2}})$ , it remains to prove the following:

(1)

$\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\mu_{\bm{c}}(p)\bm{1}_{\Delta(\bm{c})\neq 0}]=\lambda^{\natural}_{V}(p)p^{-1/2}+O(p^{-1})$ .
(2)

$\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\mu_{\bm{c}}(p^{2})\bm{1}_{\Delta(\bm{c})\neq 0}]=1+O(p^{-1})$ .
(3)

$\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\mu_{\bm{c}}(p)^{2}\bm{1}_{\Delta(\bm{c})\neq 0}]=1+O(p^{-1})$ .

We now prove (1)–(3). By [wang2023dichotomous]*Corollary 1.7 (and our definition (2.12)), we have

(6.9)	$\displaystyle\mathbb{E}_{\bm{c}\in\mathbb{F}_{p}^{m}}[E^{\natural}_{\bm{c}}(p)\bm{1}_{p\nmid\Delta(\bm{c})}]$	$\displaystyle=\lambda^{\natural}_{V}(p)p^{-1/2}+O(p^{-1}),$
(6.10)	$\displaystyle\mathbb{E}_{\bm{c}\in\mathbb{F}_{p}^{m}}[E^{\natural}_{\bm{c}}(p^{2})\bm{1}_{p\nmid\Delta(\bm{c})}]$	$\displaystyle=1+O(p^{-1}),$
(6.11)	$\displaystyle\mathbb{E}_{\bm{c}\in\mathbb{F}_{p}^{m}}[E^{\natural}_{\bm{c}}(p)^{2}\bm{1}_{p\nmid\Delta(\bm{c})}]$	$\displaystyle=1+O(p^{-1}).$

But for each $f\in\{\mu_{\bm{c}}(p),\mu_{\bm{c}}(p^{2}),\mu_{\bm{c}}(p)^{2}\}$ , we have $\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[f\bm{1}_{\Delta(\bm{c})\neq 0}]=\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[f\bm{1}_{p\nmid\Delta(\bm{c})}]+O(p^{-1})$ , by (6.8) and Lang–Weil (for $\Delta(\bm{c})\equiv 0\bmod{p}$ ). After rewriting $f\bm{1}_{p\nmid\Delta(\bm{c})}$ using (6.2), we conclude that (6.9) implies (1), that (6.11) implies (3), and that (6.10)–(6.11) imply (2). ∎

For the rest of §6, assume that $m$ is even and that Conjecture 1.2 holds.

6.2. The Sarnak–Shin–Templier framework

Using Conjecture 1.2 and (6.9)–(6.11), we can obtain useful statistical information on $L(s,V_{\bm{c}})$ over $\bm{c}\in\mathcal{S}_{1}$ .

Let $\Pi_{\bm{c}}$ be an isobaric automorphic representation over $\mathbb{Q}$ corresponding to $L(s,V_{\bm{c}})$ in Conjecture 1.2. Let $\mathcal{S}_{2}$ be the set of $\bm{c}\in\mathcal{S}_{1}$ for which $\Pi_{\bm{c}}$ is cuspidal, self-dual, and symplectic, in the sense of [sarnak2016families]*p. 533. For each $\bm{c}\in\mathcal{S}_{2}$ , the $L$ -function $L(s,V_{\bm{c}},\bigwedge^{2})$ has a pole at $s=1$ , whence there exists an isobaric automorphic representation $\phi_{\bm{c},2}$ over $\mathbb{Q}$ with

(6.12)

L(s,V_{\bm{c}},{\textstyle\bigwedge^{2}})=\zeta(s)L(s,\phi_{\bm{c},2}).

Proposition 6.2.

Let $Z\in\mathbb{R}_{\geq 1}$ . Then $\lvert(\mathcal{S}_{1}\setminus\mathcal{S}_{2})\cap[-Z,Z]^{m}\rvert\ll_{m,\epsilon}Z^{m-1/2+\epsilon}$ .

Proof.

We want to show that the family $\bm{c}\mapsto\Pi_{\bm{c}}$ indexed by $\bm{c}\in\mathcal{S}_{1}$ is essentially cuspidal, self-dual, and symplectic (in the sense of [sarnak2016families]*p. 538, (i)–(iii)), with a power-saving exceptional set. We follow the GRH strategy suggested in [sarnak2016families].

Let $\nu_{0}$ be as in §1.2. Fix $\epsilon\in(0,\frac{1}{2})$ and let $P=Z^{1-\epsilon}$ .

By Proposition 3.2(7), each $\Pi_{\bm{c}}$ is self-dual. Let $\mathcal{S}_{1.5}$ be the set of $\bm{c}\in\mathcal{S}_{1}$ for which $\Pi_{\bm{c}}$ is cuspidal. Then $L(s,V_{\bm{c}},\bigotimes^{2})$ has a pole at $s=1$ of order exactly $1$ if $\bm{c}\in\mathcal{S}_{1.5}$ , and at least $2$ if $\bm{c}\in\mathcal{S}_{1}\setminus\mathcal{S}_{1.5}$ ; this follows from the theory of unramified Rankin–Selberg $L$ -functions (cf. [farmer2019analytic]*proof of Lemma 2.3). A calculation with $\frac{L^{\prime}}{L}(s,V_{\bm{c}},\bigotimes^{2})$ (using (3.2), GRH, and [iwaniec2004analytic]*§5.6’s Exercise 6 and §5.7’s Theorem 5.15) then yields

(6.13)

\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/Z)\sum_{p\leq P:\,p\nmid\Delta(\bm{c})}(\log{p})\cdot(\lambda^{\natural}_{V_{\bm{c}},\bigotimes^{2}}(p)-1)\geq P\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/Z)(\bm{1}_{\bm{c}\notin\mathcal{S}_{1.5}}+O_{\epsilon}(P^{-1/2+\epsilon})).

On the other hand, $L(s,V_{\bm{c}},\operatorname{Sym}^{2})$ has a pole at $s=1$ if $\bm{c}\in\mathcal{S}_{1.5}\setminus\mathcal{S}_{2}$ ; this follows from [sarnak2016families]*p. 533. So a calculation with $\frac{L^{\prime}}{L}(s,V_{\bm{c}},\operatorname{Sym}^{2})$ gives

(6.14)

\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/Z)\sum_{p\leq P:\,p\nmid\Delta(\bm{c})}(\log{p})\cdot\lambda^{\natural}_{V_{\bm{c}},\operatorname{Sym}^{2}}(p)\geq P\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/Z)(\bm{1}_{\bm{c}\in\mathcal{S}_{1.5}\setminus\mathcal{S}_{2}}+O_{\epsilon}(P^{-1/2+\epsilon})).

But for all primes $p$ and tuples $\bm{c}\in\mathbb{Z}^{m}$ with $p\nmid\Delta(\bm{c})$ , we have $\bm{c}\in\mathcal{S}_{1}$ , and we can use (3.4), (3.5), (3.6) to write $\lambda^{\natural}_{V_{\bm{c}},\bigotimes^{2}}(p)=\lambda^{\natural}_{\bm{c}}(p)^{2}=E^{\natural}_{\bm{c}}(p)^{2}$ and $\lambda^{\natural}_{V_{\bm{c}},\operatorname{Sym}^{2}}(p)=\frac{1}{2}(E^{\natural}_{\bm{c}}(p)^{2}-E^{\natural}_{\bm{c}}(p^{2}))$ (since $2\nmid m_{\ast}$ ). So the left-hand side of (6.13) equals

\sum_{p\leq P}(\log{p})\sum_{\bm{c}\in\mathbb{Z}^{m}}\nu_{0}(\bm{c}/Z)(E^{\natural}_{\bm{c}}(p)^{2}-1)\bm{1}_{p\nmid\Delta(\bm{c})}=\sum_{p\leq P}(\log{p})Z^{m}(0+O(p^{-1}))\ll_{\epsilon}Z^{m}P^{\epsilon},

by Poisson summation and (6.11). Similarly, by (6.11) and (6.10), the left-hand side of (6.14) is $\ll_{\epsilon}Z^{m}P^{\epsilon}$ . Thus from (6.13) we get $\lvert(\mathcal{S}_{1}\setminus\mathcal{S}_{1.5})\cap[-Z/2,Z/2]^{m}\rvert\ll_{m,\epsilon}Z^{m}P^{-1/2+\epsilon}$ , and from (6.14) we get $\lvert(\mathcal{S}_{1.5}\setminus\mathcal{S}_{2})\cap[-Z/2,Z/2]^{m}\rvert\ll_{m,\epsilon}Z^{m}P^{-1/2+\epsilon}$ . Now let $\epsilon\to 0$ . ∎

6.3. The Ratios Recipe

By Proposition 6.1 and [sarnak2016families]*pp. 534–535, Geometric Families and Remark (i), the recipe [conrey2008autocorrelation]*§5.1 makes sense for the family $\bm{c}\mapsto\Pi_{\bm{c}}$ . We will soon derive Conjecture 1.8 accordingly, along with the following:

Conjecture 6.3 (R2 $o$ ).

Let $A_{F,2}(s_{1},s_{2})$ be defined as in §6.3.2 (in terms of $F$ ). For each real $Z\geq 2$ , let $\sigma(Z)$ be as in (1.13), and write $s_{j}=\sigma(Z)+it_{j}$ . Then there exists a real $\hbar>0$ such that uniformly over reals $Z\geq 2$ and $t_{1},t_{2}\in[-Z^{\hbar},Z^{\hbar}]$ , we have

(6.15)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\Phi^{\bm{c},1}(s_{1})\Phi^{\bm{c},1}(s_{2})=\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}(1+o_{Z\to\infty}(1))\cdot A_{F,2}(s_{1},s_{2})\zeta(s_{1}+s_{2}).

Here $A_{F,2}(s_{1},s_{2})$ is an Euler product absolutely convergent for $\operatorname{Re}(s_{1}),\operatorname{Re}(s_{2})>1/3$ .

Before proceeding, we make some remarks on our specific Ratios Conjectures.

Remark 6.4.

The $L$ -functions $L(s,V_{\bm{c}})$ are not all primitive, as the recipe in [conrey2005integral, conrey2008autocorrelation] requires. But by Proposition 6.2 and GRH, there is no real difference (in Proposition 6.1 and in our Ratios Conjectures) between $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ (on which each $L(s,V_{\bm{c}})$ is primitive).

Remark 6.5.

We do not order our families by conductor (or by discriminant, for that matter). We are indexing by different level sets, as is natural for families like ours; cf. [sarnak2016families]*p. 535, Remark (i); and p. 560, second paragraph after (25).

Remark 6.6.

In Conjectures 1.8, 1.10, and 6.3, we restrict $t$ to (comfortably) respect the constraint [conrey2007applications]*(2.11c) on “vertical shifts”. But it would be reasonable to allow $t\in[-Z^{A},Z^{A}]$ for arbitrarily large $A>0$ ; cf. [bettin2020averages]*p. 4, the sentence before Conjecture 2.

6.3.1. Deriving (RA1)

To derive Conjecture 1.8, first use (1.9) to write $\Phi^{\bm{c},1}(s)$ in terms of $1/L(s,V_{\bm{c}})$ , and then replace each term $L(s,V_{\bm{c}})^{-1}=\sum_{n\geq 1}\mu_{\bm{c}}(n)n^{-s}$ on the left-hand side of (1.14) with its “naive expected value over $\{\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\bm{c}\equiv\bm{a}\bmod{n_{0}}\}$ as $Z\to\infty$ ” (computed using Proposition 6.1), i.e. the Dirichlet series

(6.16)

\sum_{n\geq 1}\bar{\mu}_{F,1}^{\bm{a},n_{0}}(n)n^{-s}.

It turns out that the series (6.16) behaves much like $\zeta(2s)L(s+1/2,V)$ , as we now explain.

Define $A_{F,1}^{\bm{a},n_{0}}(s)$ to be the product of (6.16) and $\zeta(2s)^{-1}L(s+1/2,V)^{-1}$ , so that (6.16) factors as $A_{F,1}^{\bm{a},n_{0}}(s)\zeta(2s)L(s+1/2,V)$ . Let $\bar{a}_{F,1}^{\bm{a},n_{0}}(n)$ be the $n$ th coefficient of the Dirichlet series $A_{F,1}^{\bm{a},n_{0}}(s)$ . Then $\bar{a}_{F,1}^{\bm{a},n_{0}}(n)$ is multiplicative in $n$ by (6.3); and by (6.5), (6.6) we have

(6.17)

\bar{a}_{F,1}^{\bm{a},n_{0}}(n)\ll_{\epsilon}n^{\epsilon},\quad\bar{a}_{F,1}^{\bm{a},n_{0}}(p)\bm{1}_{p\nmid n_{0}}\ll p^{-1},\quad\bar{a}_{F,1}^{\bm{a},n_{0}}(p^{2})\bm{1}_{p\nmid n_{0}}\ll p^{-1/2}

(because $\bar{a}_{F,1}^{\bm{a},n_{0}}(p)=\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p)-\lambda^{\natural}_{V}(p)p^{-1/2}$ and $\bar{a}_{F,1}^{\bm{a},n_{0}}(p^{2})=\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p^{2})-\bar{\mu}_{F,1}^{\bm{a},n_{0}}(p)\lambda^{\natural}_{V}(p)p^{-1/2}-1+O(p^{-1})$ ). So $A_{F,1}^{\bm{a},n_{0}}(s)$ has an Euler product, and satisfies

(6.18)

\lvert A_{F,1}^{\bm{a},n_{0}}(s)\rvert\leq\sum_{n\geq 1}\frac{\lvert\bar{a}_{F,1}^{\bm{a},n_{0}}(n)\rvert}{n^{1/3+\epsilon}}\leq\prod_{p}\biggl{(}1+O(\bm{1}_{p\mid n_{0}})+\frac{O(p^{-1})}{p^{1/3+\epsilon}}+\frac{O(p^{-1/2})}{p^{2/3+2\epsilon}}+\frac{O_{\epsilon}(p^{\epsilon})}{p^{1+3\epsilon}}\biggr{)}\ll_{\epsilon}n_{0}^{\epsilon}

for $\operatorname{Re}(s)\geq 1/3+\epsilon$ (for any $\epsilon>0$ ). On the other hand, $\zeta(2s)L(s+1/2,V)$ has a pole of order $\geq 1$ at $s=1/2$ , and thus is more dominant than $A_{F,1}^{\bm{a},n_{0}}(s)$ in (6.16).

In view of the above, the Ratios Recipe [conrey2008autocorrelation]*§5.1 produces the conjecture

\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}L(s,V_{\bm{c}})^{-1}=\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}\cap Z\cdot\mathcal{B}_{M}(\bm{b}):\\ \bm{c}\equiv\bm{a}\bmod{n_{0}}\end{subarray}}(1+o_{M;Z\to\infty}(1))\cdot A_{F,1}^{\bm{a},n_{0}}(s)\zeta(2s)L(s+1/2,V)

(for $n_{0}$ , $t$ , $s$ as in Conjecture 1.8). This rearranges (upon division by $\zeta(2s)L(s+1/2,V)$ ) to (1.14), giving Conjecture 1.8. Furthermore, the fullest Ratios Conjectures include a power-saving error term, leading naturally to Conjecture 1.10.

Here $\zeta(2s)$ , $L(s+1/2,V)$ are called polar factors (or polar terms). As we will see shortly, an additional polar factor, $\zeta(s_{1}+s_{2})$ , arises in (R2).

Remark 6.7.

The Ratios Recipe involves the approximate functional equation for $L$ when there are $L$ ’s in the numerator, but not when there are only $L$ ’s in the denominator.

6.3.2. Deriving (R2)

For Conjecture 6.3, use (1.9) to write $\Phi^{\bm{c},1}(s_{j})$ in terms of $1/L(s_{j},V_{\bm{c}})$ , and then replace each term $L(s_{1},V_{\bm{c}})^{-1}L(s_{2},V_{\bm{c}})^{-1}=\sum_{n_{1},n_{2}\geq 1}\mu_{\bm{c}}(n_{1})\mu_{\bm{c}}(n_{2})n_{1}^{-s_{1}}n_{2}^{-s_{2}}$ on the left-hand side of (6.15) with its “naive expected value over $\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}$ as $Z\to\infty$ ” (computed using Proposition 6.1), i.e. the series

\sum_{n_{1},n_{2}\geq 1}\bar{\mu}_{F,2}^{\bm{0},1}(n_{1},n_{2})n_{1}^{-s_{1}}n_{2}^{-s_{2}},

which by (6.4) factors as $A_{F,2}(s_{1},s_{2})\zeta(s_{1}+s_{2})\prod_{1\leq j\leq 2}(\zeta(2s_{j})L(s_{j}+1/2,V))$ for some Euler product $A_{F,2}(s_{1},s_{2})$ . If $\operatorname{Re}(s_{1}),\operatorname{Re}(s_{2})\geq 1/3+\epsilon$ , then by (6.5) and (6.7), we have

(6.19)

\lvert A_{F,2}(s_{1},s_{2})\rvert\leq\sum_{n_{1},n_{2}\geq 1}\lvert\bar{a}_{F,2}(n_{1},n_{2})\rvert\,(n_{1}n_{2})^{-1/3-\epsilon}\ll_{\epsilon}1.

(The justification is similar to that for $A_{F,1}^{\bm{a},n_{0}}(s)$ . Note in particular that if $\bar{a}_{F,2}(n_{1},n_{2})$ is the $(n_{1},n_{2})$ th coefficient of the double Dirichlet series $A_{F,2}(s_{1},s_{2})$ , then $\bar{a}_{F,2}(p^{l},1)=\bar{a}_{F,2}(1,p^{l})=\bar{a}_{F,1}^{\bm{0},1}(p^{l})$ and $\bar{a}_{F,2}(p,p)=\bar{\mu}_{F,2}^{\bm{0},1}(p,p)-2\bar{\mu}_{F,1}^{\bm{0},1}(p)\lambda^{\natural}_{V}(p)p^{-1/2}-1+\lambda^{\natural}_{V}(p)^{2}p^{-1}$ .)

Division by $\prod_{1\leq j\leq 2}(\zeta(2s_{j})L(s_{j}+1/2,V))$ (cf. §6.3.1) leads to Conjecture 6.3.

6.4. From (R2) to (R2’) and (R2’E)

Write $A_{F,2}(\bm{s})=A_{F,2}(s_{1},s_{2})$ . Conjecture 6.3 implies, uniformly over $Z\geq 2$ and $\bm{t}\in\mathbb{R}^{2}$ , that for $s_{j}=\sigma(Z)+it_{j}$ , we have (for all $\epsilon>0$ )

(6.20)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\Phi^{\bm{c},1}(s_{1})\Phi^{\bm{c},1}(s_{2})=o_{\epsilon;Z\to\infty}(Z^{m}(1+\lVert\bm{t}\rVert)^{\epsilon})+\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}A_{F,2}(\bm{s})\zeta(s_{1}+s_{2});

this follows for $\lVert\bm{t}\rVert\leq Z^{\hbar}$ by Conjecture 6.3, and for $\lVert\bm{t}\rVert>Z^{\hbar}$ by GRH (see Proposition 3.2(8)). Using (6.20), we proceed to derive Conjecture 1.4 (R2’).

Proposition 6.8.

Assume Conjectures 1.2 and 6.3. Then Conjecture 1.4 holds.

Proof.

Let $\overline{f}(s)\colonequals\overline{f(\overline{s})}$ . Let $Z,N\geq 1$ with $N\leq Z^{3}$ . The left-hand side of (1.10) is independent of $\sigma_{0}>1/2$ , since each integrand is holomorphic on $\operatorname{Re}(s)>1/2$ by GRH. Now shift contours to $\operatorname{Re}(s)=\sigma(Z)$ , and expand squares using self-duality of the $L$ -functions in (1.9) (see Proposition 3.2(7)), to equate the left-hand side of (1.10) with the quantity

\Sigma_{0}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\int_{\sigma(Z)-i\infty}^{\sigma(Z)+i\infty}ds_{1}\int_{\sigma(Z)+i\infty}^{\sigma(Z)-i\infty}ds_{2}\,\Phi^{\bm{c},1}(s_{1})\Phi^{\bm{c},1}(s_{2})\cdot f(s_{1})\overline{f}(s_{2})N^{s_{1}+s_{2}}.

After switching the order of $\bm{c}$ and $\bm{s}$ in $\Sigma_{0}$ (using Fubini), and plugging in the estimate (6.20) and the bound $1+\lVert\bm{t}\rVert\leq(1+\lvert t_{1}\rvert)(1+\lvert t_{2}\rvert)$ for each $\bm{s}$ , we get the estimate

(6.21)

\Sigma_{0}=o_{\epsilon;Z\to\infty}(Z^{m}\Sigma_{1}^{2})+\Sigma_{2},

where $\Sigma_{1}\colonequals\int_{t\in\mathbb{R}}dt\,(1+\lvert t\rvert)^{\epsilon}\cdot\lvert f(\sigma(Z)+it)\rvert N^{\sigma(Z)}$ and

\Sigma_{2}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\int_{\sigma(Z)-i\infty}^{\sigma(Z)+i\infty}ds_{1}\int_{\sigma(Z)+i\infty}^{\sigma(Z)-i\infty}ds_{2}\,\zeta(s_{1}+s_{2})A_{F,2}(\bm{s})\cdot f(s_{1})\overline{f}(s_{2})N^{s_{1}+s_{2}}.

Here $N\leq Z^{O(1)}$ , so $\Sigma_{1}\ll N^{1/2}\sup_{0\leq\sigma\leq 2}\lVert(1+\lvert t\rvert)^{\epsilon}f(\sigma+it)\rVert_{L^{1}_{t}(\mathbb{R})}$ . And by Cauchy–Schwarz,

(6.22)

\lVert(1+\lvert t\rvert)^{\epsilon}f(\sigma+it)\rVert_{L^{1}_{t}(\mathbb{R})}\leq\lVert(1+\lvert t\rvert)^{-1/2-\epsilon}\rVert_{L^{2}_{t}(\mathbb{R})}\cdot\lVert(1+\lvert t\rvert)^{1/2+2\epsilon}f(\sigma+it)\rVert_{L^{2}_{t}(\mathbb{R})}.

Now let $\delta=1/20$ , and assume $Z$ is large enough that $1-\sigma(Z)-\delta\geq 1/3+\delta$ . Shifting $s_{2}$ (in $\Sigma_{2}$ ) from $\operatorname{Re}(s_{2})=\sigma(Z)$ to $\operatorname{Re}(s_{2})=1-\sigma(Z)-\delta$ yields $\Sigma_{2}=\Sigma_{3}+O_{F,\epsilon}(Z^{m}\Sigma_{4})$ , where

\Sigma_{3}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\int_{\sigma(Z)-i\infty}^{\sigma(Z)+i\infty}ds_{1}\,(-2\pi i)A_{F,2}(s_{1},1-s_{1})\cdot f(s_{1})\overline{f}(1-s_{1})N

comes from the residue of $\zeta(s_{1}+s_{2})$ at $s_{2}=1-s_{1}$ , and where on $\operatorname{Re}(s_{2})=1-\sigma(Z)-\delta$ we use (6.19) and the consequence $\zeta(s_{1}+s_{2})\ll_{\delta,\epsilon}(1+\lvert t_{1}+t_{2}\rvert)^{\epsilon}$ of RH to be able to take

\Sigma_{4}\colonequals\int_{\bm{t}\in\mathbb{R}^{2}}dt_{1}\,dt_{2}\,(1+\lvert t_{1}+t_{2}\rvert)^{\epsilon}\lvert f(\sigma(Z)+it_{1})f(1-\sigma(Z)-\delta-it_{2})\rvert N^{1-\delta}.

Shifting $s_{1}$ (in $\Sigma_{3}$ ) to $\operatorname{Re}(s_{1})=1/2$ yields $\Sigma_{3}\ll_{F}Z^{m}N\lVert f(1/2+it)\rVert_{L^{2}_{t}(\mathbb{R})}^{2}$ . Also, $\Sigma_{4}\ll N^{1-\delta}\sup_{0\leq\sigma\leq 2}\lVert(1+\lvert t\rvert)^{\epsilon}f(\sigma+it)\rVert_{L^{1}_{t}(\mathbb{R})}^{2}$ , since $1+\lvert t_{1}+t_{2}\rvert\leq(1+\lvert t_{1}\rvert)(1+\lvert t_{2}\rvert)$ . But $\Sigma_{0}\leq O(Z^{m}\Sigma_{1}^{2})+\Sigma_{3}+O_{F,\epsilon}(Z^{m}\Sigma_{4})$ , by (6.21). By (6.22) with $\epsilon=1/4$ , we get (1.10). ∎

As a stepping stone from Conjecture 1.4 (R2’) to Conjecture 7.4 (R2’E’), we now state (R2’E). Let $a_{\bm{c},1}(n)$ be the $n$ th coefficient of the Dirichlet series $\Phi^{\bm{c},1}(s)$ .

Conjecture 6.9 (R2’E).

Fix a function $D\in C^{\infty}_{c}(\mathbb{R}_{>0})$ . Let $t_{0}\in\mathbb{R}$ . Let $Z,N\in\mathbb{R}_{>0}$ with $N\leq Z^{3}$ . Then for some real $A_{3}>0$ depending only on $F$ , we have

(6.23)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\,\Bigl{\lvert}{\sum_{n\geq 1}D(n/N)\cdot n^{-it_{0}}a_{\bm{c},1}(n)}\Bigr{\rvert}^{2}\ll_{F,D}(1+\lvert t_{0}\rvert)^{A_{3}}Z^{m}N.

Proposition 6.10.

Assume Conjecture 1.4. Then Conjecture 6.9 holds.

Proof.

The case $N<1$ is trivial, so assume $N\geq 1$ . Let $\sigma_{0}=1.5$ . By (5.2), $D(n/N)=(2\pi)^{-1}\int_{t\in\mathbb{R}}dt\,D^{\vee}(\sigma_{0}+it)(N/n)^{\sigma_{0}+it}$ . Apply Conjecture 1.4 with $f(s)=D^{\vee}(s-it_{0})$ , and bound $f$ using Proposition 5.1, to get (6.23) with $A_{3}=2$ . ∎

6.5. From (RA1) to (RA1’E)

We now build on Conjectures 1.8 and 1.10, introducing flexible weights over $\bm{c}$ . We need some terminology on residue classes of $\mathbb{Z}^{m}$ .

Definition 6.11.

If $\mathcal{R}=\bm{a}+q\mathbb{Z}^{m}\subseteq\mathbb{Z}^{m}$ (where $q\geq 1$ ), let $q_{\mathcal{R}}\colonequals q$ be the modulus of $\mathcal{R}$ , and let $A_{F,1}^{\mathcal{R}}(s)\colonequals A_{F,1}^{\bm{a},q}(s)$ and $\bar{a}_{F,1}^{\mathcal{R}}(n)\colonequals\bar{a}_{F,1}^{\bm{a},q}(n)$ (where $A_{F,1}^{\bm{a},q}(s)$ , $\bar{a}_{F,1}^{\bm{a},q}(n)$ are defined as in §6.3.1). Given a nonempty set $\mathscr{S}=\{\mathcal{R}\}$ of residue classes $\mathcal{R}\subseteq\mathbb{Z}^{m}$ , let $Q(\mathscr{S})\colonequals\max_{\mathcal{R}\in\mathscr{S}}(q_{\mathcal{R}})$ .

Let $\mathscr{P}=\{\mathcal{R}\}$ be a partition of $\mathbb{Z}^{m}$ into finitely many residue classes $\mathcal{R}\subseteq\mathbb{Z}^{m}$ . (In §7.3, we will construct the partitions needed for our main results.) Let $I\subseteq\mathbb{R}_{>0}$ be a compact set. Let $\nu=\nu_{\bm{c}}(r)$ be a smooth function $\mathbb{R}^{m}\times\mathbb{R}\to\mathbb{C},\,(\bm{c},r)\mapsto\nu_{\bm{c}}(r)$ supported on $[-1,1]^{m}\times I$ . Let

(6.24)

\mathcal{M}_{1,k}=\mathcal{M}_{1,k}(\nu)\colonequals\sum_{\lvert\bm{\alpha}\rvert\leq 1}\,\sum_{0\leq j\leq k}\,\sup_{(\tilde{\bm{c}},r)\in\mathbb{R}^{m}\times\mathbb{R}_{>0}}\left\lvert\partial_{\tilde{\bm{c}}}^{\bm{\alpha}}\partial_{\log r}^{j}{\nu_{\tilde{\bm{c}}}(r)}\right\rvert.

Conjecture 6.12 (RA1 $o$ ’E).

Let $Z,N\geq 2Q(\mathscr{P})$ be reals with $N\leq Z^{3}$ . Then the quantity

(6.25)

\sum_{\mathcal{R}\in\mathscr{P}}\,\Bigl{\lvert}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}}\sum_{n\geq 1}\nu_{\bm{c}/Z}(n/N)\cdot(a_{\bm{c},1}(n)-\bar{a}_{F,1}^{\mathcal{R}}(n))\Bigr{\rvert}

is $\ll_{F,I}Z^{m}N^{1/2}\cdot o_{F,Q(\mathscr{P});Z\to\infty}(1)\cdot\mathcal{M}_{1,A_{4}}$ , for some real $A_{4}=A_{4}(F)>0$ .

Proposition 6.13.

Assume Conjectures 1.2, 1.4, and 1.8. Then Conjecture 6.12 holds.

Proof.

(It would be nice to prove this only assuming (1.14) for $t\in[-M,M]$ , say, but it will be convenient to assume (1.14) for all $t\in[-\log{Z},\log{Z}]$ , as in Conjecture 1.8.)

By (5.2), we have $\nu_{\bm{c}/Z}(n/N)=(2\pi i)^{-1}\int_{(\sigma(Z))}ds\,\nu_{\bm{c}/Z}^{\vee}(s)(N/n)^{s}$ , so that

\sum_{n\geq 1}\nu_{\bm{c}/Z}(n/N)(a_{\bm{c},1}(n)-\bar{a}_{F,1}^{\mathcal{R}}(n))=(2\pi i)^{-1}\int_{(\sigma(Z))}ds\,\nu_{\bm{c}/Z}^{\vee}(s)N^{s}(\Phi^{\bm{c},1}(s)-A_{F,1}^{\mathcal{R}}(s)).

Let $M\geq Q(\mathscr{P})$ be a real parameter; soon below, we will let $M$ tend slowly to infinity as $Z\to\infty$ . Recall $\mathcal{B}_{M}(\bm{b})$ from (1.12). Weight $\nu$ is supported on $[-1,1]^{m}\times I$ , so the triangle inequality and the previous display imply that (6.25) is at most

\Sigma_{5}\colonequals\sum_{\mathcal{R}\in\mathscr{P}}\sum_{\bm{b}\in[-M,M]^{m}}\,\Bigl{\lvert}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap Z\cdot\mathcal{B}_{M}(\bm{b})}\int_{(\sigma(Z))}ds\,\nu_{\bm{c}/Z}^{\vee}(s)N^{s}(\Phi^{\bm{c},1}(s)-A_{F,1}^{\mathcal{R}}(s))\Bigr{\rvert}.

Uniformly over $\bm{c}\in\mathbb{R}^{m}$ and $s\in\mathbb{C}$ , Proposition 5.1 and (6.24) give (for all $b\in\mathbb{Z}_{\geq 0}$ )

(6.26)

\nu_{\bm{c}/Z}^{\vee}(s)\ll_{b}O_{I}(1)^{1+\lvert\operatorname{Re}(s)\rvert}\mathcal{M}_{1,b}(\nu)(1+\lvert\operatorname{Im}(s)\rvert)^{-b}.

For each pair $(\mathcal{R},\bm{b})$ , choose an element $\bm{c}(\mathcal{R},\bm{b})$ of $\mathcal{S}_{1}\cap\mathcal{R}\cap Z\cdot\mathcal{B}_{M}(\bm{b})$ , if such an element exists. Let $\Sigma_{6}$ be $\Sigma_{5}$ with $\nu_{\bm{c}/Z}^{\vee}(s)-\nu_{\bm{c}(\mathcal{R},\bm{b})/Z}^{\vee}(s)$ in place of $\nu_{\bm{c}/Z}^{\vee}(s)$ , and let $\Sigma_{7}$ be $\Sigma_{5}$ with $\nu_{\bm{c}(\mathcal{R},\bm{b})/Z}^{\vee}(s)$ in place of $\nu_{\bm{c}/Z}^{\vee}(s)$ . Clearly $\Sigma_{5}\leq\Sigma_{6}+\Sigma_{7}$ .

We need to split $\Sigma_{7}$ further according to the size of $t$ , with some analytic care (keeping in mind the entireness hypothesis on $f$ in Conjecture 1.4). Let $B=B_{M}(s)\colonequals e^{(s/M)^{2}}$ . The following hold uniformly over $\sigma$ in any fixed finite interval:

(1)

For all $t\in\mathbb{R}$ , we have $B(\sigma+it)\ll 1$ .
(2)

If $\lvert t\rvert>\log{Z}$ , then $B(\sigma+it)\ll_{M}Z^{-1}$ .
(3)

If $\lvert t\rvert\leq M^{1/2}$ , then $B(\sigma+it)=1+O(M^{-1})$ .

Let $\Sigma_{7}(A)$ be $\Sigma_{5}$ with $\nu_{\bm{c}(\mathcal{R},\bm{b})/Z}^{\vee}(s)\cdot A(s)$ in place of $\nu_{\bm{c}/Z}^{\vee}(s)$ , so that

(6.27)

\Sigma_{7}\leq\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert\leq\log{Z}})+\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert>\log{Z}})+\Sigma_{7}(1-B).

We first bound $\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert\leq\log{Z}})$ and $\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert>\log{Z}})$ . Plugging (1), (6.26) (with $b=2$ ), and Conjecture 1.8 (for $t\in[-\log{Z},\log{Z}]$ ) into $\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert\leq\log{Z}})$ , we find that

\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert\leq\log{Z}})\leq\sum_{\mathcal{R}\in\mathscr{P}}\sum_{\bm{b}\in[-M,M]^{m}}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap Z\cdot\mathcal{B}_{M}(\bm{b})}O_{I}(\mathcal{M}_{1,2}(\nu))N^{\sigma(Z)}o_{M;Z\to\infty}(1),

which is in turn $\ll_{I}Z^{m}N^{1/2}o_{M;Z\to\infty}(1)\mathcal{M}_{1,2}(\nu)$ . Similarly, plugging (2), (6.26) (with $b=2$ ), GRH (see Proposition 3.2(8)), and (6.18) into $\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert>\log{Z}})$ reveals that

\Sigma_{7}(B\cdot\bm{1}_{\lvert t\rvert>\log{Z}})\ll_{M,\epsilon}\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}Z^{-1}O_{I}(\mathcal{M}_{1,2}(\nu))N^{\sigma(Z)}(Z^{\epsilon}+Q(\mathscr{P})^{\epsilon}),

which (if $\epsilon=1/2$ , say) is $\ll_{I}Z^{m}N^{1/2}o_{M;Z\to\infty}(1)\mathcal{M}_{1,2}(\nu)$ (since $Q(\mathscr{P})\leq Z$ ).

By choosing $M=M(Z)\geq Q(\mathscr{P})$ appropriately, we may ensure both (i) that $M\to\infty$ as $Z\to\infty$ , and (ii) that the two “ $o_{M;Z\to\infty}(1)$ ” terms in the previous paragraph are $o_{Z\to\infty}(1)$ .

It remains to bound $\Sigma_{6}$ and $\Sigma_{7}(1-B)$ . To handle both at once, we need a Fourier analog of Lemma 5.2. Let $f\colon\mathbb{C}\to\mathbb{C}$ be one of the functions $f_{\bm{c},6}\colon s\mapsto\nu_{\bm{c}/Z}^{\vee}(s)-\nu_{\bm{c}(\mathcal{R},\bm{b})/Z}^{\vee}(s)$ (given $(\mathcal{R},\bm{b},\bm{c})$ with $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap Z\cdot\mathcal{B}_{M}(\bm{b})$ ), $f_{\bm{c},7}\colon s\mapsto\nu_{\bm{c}/Z}^{\vee}(s)\cdot(1-B(s))$ (given $\bm{c}\in\mathcal{S}_{1}$ ). Let

(6.28)

g_{t_{0}}(t)=f(it)\nu_{2}^{\vee}(-i(t-t_{0}))/2\pi.

Note that $f$ , $\nu_{2}^{\vee}$ are entire (and rapidly decaying in vertical strips), so $g_{t_{0}}$ is entire (and rapidly decaying in horizontal strips). By Parseval’s theorem and (5.4), we have

\int_{\mathbb{R}}dt\,\nu_{2}^{\vee}(it)\nu_{2}^{\vee}(-it)=2\pi\int_{r>0}d^{\times}{r}\,\nu_{2}(r)^{2}=2\pi.

So $f(it)=\int_{\mathbb{R}}dt_{0}\,g_{t_{0}}(t)\nu_{2}^{\vee}(i(t-t_{0}))$ by (6.28). By Fourier inversion applied to $g_{t_{0}}$ , we get

f(it)=\int_{\mathbb{R}^{2}}dt_{0}\,dx\,\hat{g}_{t_{0}}(x)e(xt)\nu_{2}^{\vee}(i(t-t_{0}))=\int_{\mathbb{R}}dt_{0}\int_{y>0}d^{\times}{y}\,\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})y^{it}\nu_{2}^{\vee}(i(t-t_{0})).

By analytic continuation, it follows that for any $\sigma>1/2$ , the quantity

(6.29)

\int_{(\sigma)}ds\,f(s)N^{s}(\Phi^{\bm{c},1}(s)-A_{F,1}^{\mathcal{R}}(s))

equals $\int_{\mathbb{R}}dt_{0}\int_{y>0}d^{\times}{y}\,\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})\int_{(\sigma)}ds\,y^{s}\nu_{2}^{\vee}(s-it_{0})N^{s}(\Phi^{\bm{c},1}(s)-A_{F,1}^{\mathcal{R}}(s))$ , and thus has absolute value at most

(6.30)

\int_{\mathbb{R}}dt_{0}\int_{y>0}d^{\times}{y}\,\lvert\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})\rvert\Bigl{\lvert}{\int_{(\sigma)}ds\,\nu_{2}^{\vee}(s-it_{0})(Ny)^{s}(\Phi^{\bm{c},1}(s)-A_{F,1}^{\mathcal{R}}(s))}\Bigr{\rvert}.

But for all $z\in\mathbb{C}$ , we have $\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})=\int_{\mathbb{R}}dt\,g_{t_{0}}(t)y^{-it}=\int_{\mathbb{R}}dt\,g_{t_{0}}(t+z)y^{-i(t+z)}$ (by the definition of $\hat{g}_{t_{0}}$ , if $z=0$ ; and then by analytic continuation, in general), and thus $\lvert\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})\rvert\leq y^{\operatorname{Im}(z)}\int_{\mathbb{R}}dt\,\lvert g_{t_{0}}(t+z)\rvert$ . Using (6.28), Proposition 5.1 (cf. (6.26)), and the definitions of $\mathcal{M}_{1,b}$ (see (6.24)) to bound $\lvert g_{t_{0}}(t+z)\rvert$ pointwise, we then get (by taking $z=\pm iu\in i\mathbb{R}$ )

(6.31)

\hat{g}_{t_{0}}(\tfrac{\log{y}}{2\pi})\ll_{u,I,b,\nu_{2},B}M^{-1}\mathcal{M}_{1,b}(\nu)(1+\lvert t_{0}\rvert)^{-b}(y^{u}+y^{-u})^{-1}

for all reals $u\geq 0$ and integers $b\geq 0$ . (If $f=f_{\bm{c},6}$ , the factor of $M^{-1}$ in (6.31) arises from the bound $\lVert\bm{c}-\bm{c}(\mathcal{R},\bm{b})\rVert\ll Z/M$ . If $f=f_{\bm{c},7}$ , the factor of $M^{-1}$ comes from (3) over $\lvert t\rvert\leq M^{1/2}$ , and from the decay of $g_{t_{0}}(t)$ in $t$ , $t-t_{0}$ if $\lvert t\rvert>M^{1/2}$ .)

Note that for all $\sigma\geq 1/3+\epsilon$ , we may apply (6.18) and Proposition 5.1 (after shifting contours to $\operatorname{Re}(s)=1/3+\epsilon$ ) to get (provided $0<\epsilon\leq 1/12$ )

(6.32)

\int_{(\sigma)}ds\,\nu_{2}^{\vee}(s-it_{0})(Ny)^{s}A_{F,1}^{\mathcal{R}}(s)\ll_{\epsilon}\int_{t\in\mathbb{R}}dt\,\frac{(Ny)^{1/3+\epsilon}Q(\mathscr{P})^{\epsilon}}{(1+\lvert t-t_{0}\rvert)^{2}}\ll(Ny+Q(\mathscr{P}))^{1/2}.

In view of the bound (6.30) for (6.29), we may now apply (6.31), (6.32), Conjecture 1.4 (with $Z+(Ny)^{1/3}$ , $Ny$ , $\nu_{2}^{\vee}(s-it_{0})$ in place of $Z$ , $N$ , $f(s)$ ), and Cauchy–Schwarz to get

\Sigma_{6}\ll_{I}\int_{\mathbb{R}}dt_{0}\int_{y>0}d^{\times}{y}\,\frac{M^{-1}\mathcal{M}_{1,b}(\nu)}{(1+\lvert t_{0}\rvert)^{b}(y^{u}+y^{-u})}(Z+(Ny)^{1/3})^{m}(Ny+Q(\mathscr{P}))^{1/2}(1+\lvert t_{0}\rvert),

which is $\ll_{I}M^{-1}\mathcal{M}_{1,b}(\nu)Z^{m}N^{1/2}$ by Lemma 5.3 (provided $u\geq m/3+1$ and $b\geq 3$ , and $N\geq Q(\mathscr{P})$ ). The same holds for $\Sigma_{7}(1-B)$ . Thus Conjecture 6.12 holds with $A_{4}=3$ . ∎

Proposition 6.14 (RA1 $\delta$ ’E).

Assume Conjectures 1.2 and 1.10. Suppose $Z,N\geq 2Q(\mathscr{P})$ with $N\leq Z^{3}$ and $Q(\mathscr{P})\leq Z^{\eta_{2}}$ . Then the quantity (6.25) is $\ll_{F,I}Z^{m-\eta_{2}}N^{1/2}\mathcal{M}_{1,A_{5}}$ . Here $\eta_{2}$ , $A_{5}$ are positive reals depending only on $F$ .

Proof.

Mimic the proof of Proposition 6.13, but take $M=Z^{\eta_{1}}$ , replace Conjecture 1.8 with 1.10, replace every use of Conjecture 1.4 with GRH, and replace (6.27) with the bound $\Sigma_{7}\leq\Sigma_{7}(\bm{1}_{\lvert t\rvert\leq Z^{\eta_{1}}})+\Sigma_{7}(\bm{1}_{\lvert t\rvert>Z^{\eta_{1}}})$ . The details simplify, since the desired final bound is not sensitive to losses of $Z^{\epsilon}$ . (We can take $\eta_{2}=\eta_{1}-\epsilon$ and $A_{5}=3$ .) ∎

When applying Propositions 6.13 and 6.14 (in §7.4), we need the following lemma:

Lemma 6.15.

Fix a real $\theta>0$ . Suppose $\mathcal{S}\subseteq\mathbb{Z}^{m}$ and $\lvert\mathcal{S}\cap[-C,C]^{m}\rvert\ll C^{\theta}$ for all reals $C\geq 1$ . Let $Z,N\geq 1$ be reals. Then

(6.33)

\sum_{\mathcal{R}\in\mathscr{P}}\,\Bigl{\lvert}\sum_{\bm{c}\in\mathcal{S}\cap\mathcal{R}}\sum_{n\geq 1}\nu_{\bm{c}/Z}(n/N)\bar{a}_{F,1}^{\mathcal{R}}(n)\Bigr{\rvert}\ll_{F,I,\theta,\epsilon}Z^{\theta}N^{1/3+\epsilon}Q(\mathscr{P})^{\epsilon}\mathcal{M}_{1,0}.

Proof.

By the triangle inequality, the left-hand side of (6.33) is at most

\sum_{\mathcal{R}\in\mathscr{P}}\sum_{\bm{c}\in\mathcal{S}\cap\mathcal{R}}\left(\sup_{r>0}{\lvert\nu_{\bm{c}/Z}(r)\rvert}\right)\sum_{n\in N\cdot I}\lvert\bar{a}_{F,1}^{\mathcal{R}}(n)\rvert\leq\biggl{(}\,\sup_{\mathcal{R}\in\mathscr{P}}\sum_{n\in N\cdot I}\lvert\bar{a}_{F,1}^{\mathcal{R}}(n)\rvert\biggr{)}\sum_{\bm{c}\in\mathcal{S}}\sup_{r>0}{\lvert\nu_{\bm{c}/Z}(r)\rvert},

since $\nu$ is supported on $\mathbb{R}^{m}\times I$ and $\mathscr{P}$ is a partition of $\mathbb{Z}^{m}$ . But $\sup_{\bm{c}\in\mathbb{R}^{m}}\sup_{r>0}{\lvert\nu_{\bm{c}/Z}(r)\rvert}\leq\mathcal{M}_{1,0}(\nu)$ by (6.24); so $\sum_{\bm{c}\in\mathcal{S}}\sup_{r>0}{\lvert\nu_{\bm{c}/Z}(r)\rvert}\ll_{\theta}Z^{\theta}\mathcal{M}_{1,0}(\nu)$ (since $\operatorname{Supp}{\nu}\subseteq[-1,1]^{m}\times I$ ). Also, $\sup_{\mathcal{R}\in\mathscr{P}}\sum_{n\in N\cdot I}\lvert\bar{a}_{F,1}^{\mathcal{R}}(n)\rvert\ll_{I,\epsilon}Q(\mathscr{P})^{\epsilon}N^{1/3+\epsilon}$ by (6.18). Multiplying gives (6.33). ∎

6.6. Bounding exterior squares

For each $\bm{c}\in\mathcal{S}_{1}$ , let $\mu_{\bm{c},2}(n)$ denote the $n$ th coefficient of the Dirichlet series $\zeta(s)/L(s,V_{\bm{c}},\bigwedge^{2})$ . Recall $\mathcal{N}^{\bm{c}}$ , $\mathcal{N}_{\bm{c}}$ from (2.2). Assume Conjecture 1.2.

Proposition 6.16.

Let $A\geq 2$ be an even integer. Let $Z,N,\epsilon\in\mathbb{R}_{>0}$ with $N\leq Z^{3/2}$ . Then

(6.34)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\,\Bigl{\lvert}{\sum_{N\leq n<2N:\,n\in\mathcal{N}^{\bm{c}}}\mu_{\bm{c},2}(n)}\Bigr{\rvert}^{A}\ll_{A,\epsilon}Z^{m}N^{A-1/3+\epsilon}.

Proof.

For $N<2$ , the left-hand side of (6.34) is $\ll_{A}Z^{m}$ by (3.2). Now assume $N\geq 2$ .

Let $\nu_{0}$ be as in §1.2. Let $f(Z,N)$ denote the left-hand side of (6.34). Let $g(Z,N)$ denote $f(Z,N)$ with $\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/Z)\cdots$ in place of $\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\cdots$ . Then by positivity,

(6.35)

g(Z/2,N)\leq f(Z,N)\leq g(2Z,N)

for all $Z>0$ . But if $Z\geq(2N)^{A+1}$ , then (if we let $\mu_{\bm{c},2}(\bm{n})\colonequals\mu_{\bm{c},2}(n_{1})\cdots\mu_{\bm{c},2}(n_{A})$ and $P\colonequals n_{1}\cdots n_{A}$ , and note that $\bm{1}_{\gcd(P,\Delta(\bm{c}))=1}\cdot\mu_{\bm{c},2}(\bm{n})$ is determined by $P$ and $\bm{c}\bmod{P}$ )

\begin{split}g(Z,N)&=\sum_{N\leq n_{1},\dots,n_{A}<2N}\sum_{\bm{c}\in\mathbb{Z}^{m}}\nu_{0}(\bm{c}/Z)\bm{1}_{\bm{c}\in\mathcal{S}_{1}}\bm{1}_{\gcd(P,\Delta(\bm{c}))=1}\cdot\mu_{\bm{c},2}(\bm{n})\\ &=O_{A}(Z^{m})+\sum_{N\leq n_{1},\dots,n_{A}<2N}(Z/P)^{m}\sum_{\bm{c}\in(\mathbb{Z}/P\mathbb{Z})^{m}}\bm{1}_{\gcd(P,\Delta(\bm{c}))=1}\cdot\mu_{\bm{c},2}(\bm{n}),\end{split}

by (3.2) (or (3.8)) and Poisson summation in residue classes modulo $P$ (cf. the proof of Proposition 6.2); note that $P\geq N^{A}>1$ , so the condition $\gcd(P,\Delta(\bm{c}))=1$ automatically implies $\bm{c}\in\mathcal{S}_{1}$ . If $\tilde{g}(Z,N)\colonequals g(Z,N)/Z^{m}$ , then for $Z\geq(2N)^{A+1}$ we conclude that

(6.36)

\tilde{g}(Z,N)=\tilde{g}((2N)^{A+1},N)+O_{A}(1).

We now address $Z\leq(2N)^{A+1}$ . Recall $\mathcal{S}_{2}$ from Proposition 6.2. If $\bm{c}\in\mathcal{S}_{2}$ , then (6.12) and Proposition 3.2(8) (applied to $L(s,\phi_{\bm{c},2})^{-1}$ ), when combined with (3.2) at primes $p\mid\Delta(\bm{c})$ , yield $\sum_{N\leq n<2N:\,n\in\mathcal{N}^{\bm{c}}}\mu_{\bm{c},2}(n)\ll_{\epsilon}\lVert\bm{c}\rVert^{\epsilon}N^{1/2+\epsilon}$ . If $\bm{c}\in\mathcal{S}_{1}\setminus\mathcal{S}_{2}$ , we still have the bound $\sum_{N\leq n<2N:\,n\in\mathcal{N}^{\bm{c}}}\mu_{\bm{c},2}(n)\ll_{\epsilon}N^{1+\epsilon}$ due to (3.2). Therefore, Proposition 6.2 yields

g(Z,N)\ll_{A,\epsilon}Z^{m+\epsilon}N^{A/2+\epsilon}+Z^{m-1/2+\epsilon}N^{A+\epsilon}

for all $Z>0$ . It follows that $\tilde{g}(Z,N)\ll_{A,\epsilon}N^{A/2+\epsilon}+N^{A-1/3+\epsilon}$ for $Z\geq N^{2/3}$ when $Z\leq(2N)^{A+1}$ , and thus (by (6.36)) for all $Z\geq N^{2/3}$ . Now (6.34) follows from (6.35). ∎

To handle primes $p\mid\Delta(\bm{c})$ , we prove the following (unconditional) result:

Lemma 6.17.

If $Z,A,\epsilon\in\mathbb{R}_{>0}$ , then $\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[1,N]\rvert^{A}\ll_{A,\epsilon}Z^{m}N^{\epsilon}$ .

Proof.

Lemma 2.1 immediately suffices if $Z<N^{A}$ . Now suppose $Z\geq N^{A}$ . By Hölder’s inequality, we may assume $A\in\mathbb{Z}_{\geq 1}$ . The sum $\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[1,N]\rvert^{A}$ then equals

\Sigma_{8}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\sum_{u_{1},\dots,u_{A}\leq N:\,u_{i}\mid\Delta(\bm{c})^{\infty}}1=\sum_{u_{1},\dots,u_{A}\leq N}\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\bm{1}_{\operatorname{rad}(u_{1}\cdots u_{A})\mid\Delta(\bm{c})}.

In $\Sigma_{8}$ we have $u_{1}\cdots u_{A}\leq N^{A}\leq Z$ , so by Lang–Weil and the Chinese remainder theorem,

\Sigma_{8}\ll_{\epsilon}\sum_{u_{1},\dots,u_{A}\leq N}Z^{m}\operatorname{rad}(u_{1}\cdots u_{A})^{\epsilon-1}\leq\sum_{r\leq N^{A}}Z^{m}r^{\epsilon-1}\sum_{u_{1},\dots,u_{A}\leq N:\,u_{i}\mid r^{\infty}}1.

By Lemma 2.1, we conclude that $\Sigma_{8}\ll_{A,\epsilon}\sum_{r\leq N^{A}}Z^{m}r^{\epsilon-1}(Nr)^{\epsilon}\ll_{\epsilon}Z^{m}N^{(2A+1)\epsilon}$ . ∎

In §7, we need the following technical complement to Proposition 6.10.

Proposition 6.18 ( $\bigwedge$ 2E).

Assume Conjecture 1.2. Fix $A\in\mathbb{R}_{>0}$ and $f(r)\in\{\bm{1}_{r\leq 1}\}\cup C^{\infty}_{c}(\mathbb{R}_{>0})$ . Let $Z,N\in\mathbb{R}_{>0}$ with $N\leq Z^{3/2}$ . Let $t_{0}\in\mathbb{R}$ . For some $\eta_{3}=\eta_{3}(A)>0$ , we have

(6.37)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\,\Bigl{\lvert}{\sum_{n\geq 1}f(n/N)n^{-it_{0}}\cdot\mu_{\bm{c},2}(n)}\Bigr{\rvert}^{A}\ll_{A,f}(1+\lvert t_{0}\rvert)^{A}Z^{m}N^{(1-\eta_{3})A}.

Proof.

First suppose $(f,t_{0})=(\bm{1}_{r\leq 1},0)$ and $A\in\mathbb{Z}_{\geq 2}$ . Let $g(M)=\sum_{r\leq M:\,r\in\mathcal{N}^{\bm{c}}}\mu_{\bm{c},2}(r)$ . By multiplicativity, $\sum_{n\leq N}\mu_{\bm{c},2}(n)=\sum_{d\leq N:\,d\in\mathcal{N}_{\bm{c}}}\mu_{\bm{c},2}(d)g(N/d)$ , which is $\ll_{\epsilon}N^{\epsilon}\sum_{d\leq N:\,d\in\mathcal{N}_{\bm{c}}}\lvert g(N/d)\rvert$ by (3.2). So by Hölder over $d$ , the left-hand side of (6.37) is at most $O_{A,\epsilon}(N^{\epsilon})$ times

\Sigma_{9}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[1,N]\rvert^{A-1}\sum_{d\leq N:\,d\in\mathcal{N}_{\bm{c}}}\lvert g(N/d)\rvert^{A}.

Switching $\bm{c}$ , $d$ yields $\Sigma_{9}\leq\sum_{d\leq N}\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[1,N]\rvert^{A-1}\lvert g(N/d)\rvert^{A}$ (by positivity). Then Cauchy–Schwarz over $\bm{c}$ , followed by Lemma 6.17 and Proposition 6.16, gives

\Sigma_{9}\ll_{A,\epsilon}\sum_{d\leq N}(Z^{m}N^{\epsilon})^{1/2}(Z^{m}(N/d)^{2A-1/3+\epsilon})^{1/2}\ll_{A,\epsilon}Z^{m}N^{A-1/6+\epsilon},

where we use $A-1/6+\epsilon/2\geq 1+\epsilon/2$ to evaluate the sum over $d$ . By Hölder, then, (6.37) holds with $\eta_{3}=1/7A$ if $A\geq 2$ , and with $\eta_{3}=1/14$ if $0<A\leq 2$ . The general case follows from partial summation and Hölder, since $\frac{\partial}{\partial n}D(n/N)\ll_{D}1/N$ and $\frac{\partial}{\partial n}n^{it_{0}}\ll\lvert t_{0}\rvert/n$ . ∎

(With more work, one could relax the GRH assumption. One could in fact unconditionally handle the case of very large $Z$ , using (6.9)–(6.11); see [wang2021_HLH_vs_RMT]*§7.6.)

7. Adapting $L$ -function statistics to delta

7.1. Factorization

We need to mold the statistics from §6 into a form friendlier for the delta method. We first split the series $\Phi(\bm{c},s)$ (from (2.11)) into more manageable pieces. Recall $\mathcal{N}^{\bm{c}}$ , $\mathcal{N}_{\bm{c}}$ from (2.2). Given $\bm{c}\in\mathcal{S}_{1}$ , consider the factorization $\Phi=\Phi^{\operatorname{G}}\Phi^{\operatorname{B}}$ , where

(7.1)		$\displaystyle\Phi^{\operatorname{G}}(\bm{c},s)$	$\displaystyle\colonequals\prod_{p\nmid\Delta(\bm{c})}\Phi_{p}(\bm{c},s)=\sum_{n\in\mathcal{N}^{\bm{c}}}S^{\natural}_{\bm{c}}(n)n^{-s},$
(7.2)		$\displaystyle\Phi^{\operatorname{B}}(\bm{c},s)$	$\displaystyle\colonequals\prod_{p\mid\Delta(\bm{c})}\Phi_{p}(\bm{c},s)=\sum_{n\in\mathcal{N}_{\bm{c}}}S^{\natural}_{\bm{c}}(n)n^{-s}.$

One can approximate $\Phi^{\operatorname{G}}$ using Hasse–Weil $L$ -functions. It would be nice to also relate $\Phi_{p}$ for $p\mid\Delta(\bm{c})$ to $L$ -functions, even in special cases like when $m=4$ and $v_{p}(\Delta(\bm{c}))=1$ . For now, we study $\Phi^{\operatorname{B}}$ by completely different means (see §9). In §7, we thus concentrate on $\Phi^{\operatorname{G}}$ .

For the rest of §7, assume $2\mid m$ . We first factor $\Phi^{\operatorname{G}}$ into three pieces: $\Phi^{\bm{c},1}$ , $\Phi^{\bm{c},2}$ , $\Phi^{\bm{c},3}$ .

Definition 7.1.

Let $\Phi^{\bm{c},1}(s)\colonequals L(s,V_{\bm{c}})^{-1}L(1/2+s,V)^{-1}\zeta(2s)^{-1}$ as in (1.9), and let

(7.3)		$\displaystyle\Phi^{\bm{c},2}(s)$	$\displaystyle\colonequals\zeta(2s)/L(2s,V_{\bm{c}},\textstyle{\bigwedge^{2}}),$
(7.4)		$\displaystyle\Phi^{\bm{c},3}(s)$	$\displaystyle\colonequals\Phi^{\operatorname{G}}(\bm{c},s)L(s,V_{\bm{c}})L(1/2+s,V)L(2s,V_{\bm{c}},\textstyle{\bigwedge^{2}}).$

For each $j\in\{1,2,3\}$ , let $a_{\bm{c},j}(n)$ be the $n$ th coefficient of the Dirichlet series $\Phi^{\bm{c},j}(s)$ .

The factors $\Phi^{\bm{c},2}$ , $\Phi^{\bm{c},3}$ in $\Phi^{\operatorname{G}}=\Phi^{\bm{c},1}\Phi^{\bm{c},2}\Phi^{\bm{c},3}$ hinder any attempt to apply statistics on $\Phi^{\bm{c},1}$ to $\Phi^{\operatorname{G}}$ . Fortunately, $\Phi^{\bm{c},2}$ , $\Phi^{\bm{c},3}$ turn out to behave as “error factors” on average. Proposition 6.18 lets us handle large moduli in $\Phi^{\bm{c},2}$ ; note that $a_{\bm{c},2}(n)=0$ if $n$ is not a square, and

(7.5)

a_{\bm{c},2}(n)=\mu_{\bm{c},2}(n^{1/2})

otherwise (where $\mu_{\bm{c},2}$ is as in §6.6). We now prove results to handle large moduli in $\Phi^{\bm{c},3}$ .

The factor $\Phi^{\bm{c},3}$ measures the quality of $\Phi^{\bm{c},1}\Phi^{\bm{c},2}$ as an approximation to $\Phi^{\operatorname{G}}$ . Recall the “first-order approximation” $\Phi(\bm{c},s)=\Psi^{\bm{c},1}(s)\Psi^{\bm{c},2}(s)$ given by $\Psi^{\bm{c},1}$ , $\Psi^{\bm{c},2}$ from (2.18); here $\Psi^{\bm{c},1}(s)=1/L(s,V_{\bm{c}})$ . The “first-order error” $\Psi^{\bm{c},2}(s)$ is only expected to converge absolutely for $\operatorname{Re}(s)>1/2$ . As suggested in §2, this is a “source of $\epsilon$ ” in (2.17). On the other hand, the following result establishes absolute convergence for $\Phi^{\bm{c},3}(s)$ past the critical line $\operatorname{Re}(s)=1/2$ .

Proposition 7.2.

Uniformly over $\bm{c}\in\mathcal{S}_{1}$ , primes $p$ , and integers $l\geq 1$ , we have

(7.6)

a_{\bm{c},3}(p)\cdot\bm{1}_{p\nmid\Delta(\bm{c})}=0,\quad a_{\bm{c},3}(p^{2})\cdot\bm{1}_{p\nmid\Delta(\bm{c})}\ll p^{-1/2},\quad a_{\bm{c},3}(p^{l})\ll_{\epsilon}p^{l\epsilon}.

In particular, if $\bm{c}\in\mathcal{S}_{1}$ , then $\Phi^{\bm{c},3}(s)$ converges absolutely over $\operatorname{Re}(s)>1/3$ .

Proof.

Let $\bm{c}\in\mathcal{S}_{1}$ . Let $\lambda^{\natural}_{\bm{c}}\colonequals\lambda^{\natural}_{V_{\bm{c}}}$ , as in §6.1. Suppose first that $p\mid\Delta(\bm{c})$ . Then $\Phi^{\operatorname{G}}_{p}(\bm{c},s)=1$ by (7.1). So by (7.4) and (3.2), we have $a_{\bm{c},3}(p^{l})\ll_{\epsilon}p^{l\epsilon}$ . So (7.6) holds.

Now suppose $p\nmid\Delta(\bm{c})$ . Then $\Phi_{p}(\bm{c},s)=1+S^{\natural}_{\bm{c}}(p)p^{-s}$ by (2.14), and

S^{\natural}_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p)-p^{-1/2}E^{\natural}_{F}(p)=-\lambda^{\natural}_{\bm{c}}(p)-p^{-1/2}\lambda^{\natural}_{V}(p)

by (2.13), (3.4), and (3.3). In particular, $a_{\bm{c},3}(p^{l})\ll_{\epsilon}p^{l\epsilon}$ by (7.4) and (3.2). Furthermore,

\begin{split}\Phi_{p}(\bm{c},s)L_{p}(s,V_{\bm{c}})&=(1-\lambda^{\natural}_{\bm{c}}(p)p^{-s}-\lambda^{\natural}_{V}(p)p^{-1/2-s})(1+\lambda^{\natural}_{\bm{c}}(p)p^{-s}+\lambda^{\natural}_{\bm{c}}(p^{2})p^{-2s}+O(p^{-3s}))\\ &=1-\lambda^{\natural}_{V}(p)p^{-1/2-s}+[\lambda^{\natural}_{\bm{c}}(p^{2})-\lambda^{\natural}_{\bm{c}}(p)^{2}]p^{-2s}+O(p^{-1/2-2s})+O(p^{-3s}).\end{split}

To get further cancellation, we multiply $\Phi_{p}(\bm{c},s)L_{p}(s,V_{\bm{c}})$ by

\begin{split}L_{p}(1/2+s,V)&=1+\lambda^{\natural}_{V}(p)p^{-1/2-s}+O(p^{-1-2s}),\\ L_{p}(2s,V_{\bm{c}},\textstyle{\bigwedge^{2}})&=1+\lambda^{\natural}_{V_{\bm{c}},\bigwedge^{2}}(p)p^{-2s}+O(p^{-4s}),\end{split}

to get (in view of $\lambda^{\natural}_{V_{\bm{c}},\bigwedge^{2}}(p)=\lambda^{\natural}_{\bm{c}}(p)^{2}-\lambda^{\natural}_{\bm{c}}(p^{2})$ from (3.6))

(7.7)

\Phi_{p}(\bm{c},s)L_{p}(s,V_{\bm{c}})L_{p}(1/2+s,V)L_{p}(2s,V_{\bm{c}},\textstyle{\bigwedge^{2}})=1+O(p^{-1/2-2s})+O(p^{-3s}).

By (7.4), the left-hand side of (7.7) is precisely the local factor $\Phi^{\bm{c},3}_{p}(s)$ of $\Phi^{\bm{c},3}(s)$ . Thus (7.7) completes the proof of (7.6). The convergence statement on $\Phi^{\bm{c},3}(s)$ follows from (7.6). ∎

Corollary 7.3 ( $\Phi$ 3E).

Fix $A\in\mathbb{R}_{>0}$ . Then uniformly over $Z,N\in\mathbb{R}_{>0}$ , we have

(7.8)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\biggl{(}\,\sum_{n\leq N}\lvert a_{\bm{c},3}(n)\rvert\biggr{)}^{\!A}\ll_{A,\epsilon}Z^{m}N^{(1/3+\epsilon)A}.

Proof.

Recall $\mathcal{N}^{\bm{c}}$ , $\mathcal{N}_{\bm{c}}$ from (2.2). By multiplicativity (of $\lvert a_{\bm{c},3}\rvert$ ) and positivity, we have

(7.9)

\sum_{n\leq N}\lvert a_{\bm{c},3}(n)\rvert\leq\sum_{d\leq N:\,d\in\mathcal{N}_{\bm{c}}}\lvert a_{\bm{c},3}(d)\rvert\sum_{r\leq N:\,r\in\mathcal{N}^{\bm{c}}}\lvert a_{\bm{c},3}(r)\rvert.

But for any $\epsilon>0$ , the bounds in (7.6) imply $a_{\bm{c},3}(d)\ll_{\epsilon}d^{\epsilon}$ , and that $\sum_{r\leq N:\,r\in\mathcal{N}^{\bm{c}}}\lvert a_{\bm{c},3}(r)\rvert$ is $\leq N^{1/3+\epsilon}\sum_{r\in\mathcal{N}^{\bm{c}}}r^{-1/3-\epsilon}\lvert a_{\bm{c},3}(r)\rvert\ll_{\epsilon}N^{1/3+\epsilon}$ (cf. (6.18)). So by (7.9), the left-hand side of (7.8) is $\ll_{A,\epsilon}N^{(1/3+2\epsilon)A}\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[1,N]\rvert^{A-1}$ . Now (7.8) follows from Lemma 6.17. ∎

7.2. From (R2’E) to (R2’E’)

We now build on Conjecture 6.9 (R2’E).

Conjecture 7.4 (R2’E’).

Fix a function $D\in C^{\infty}_{c}(\mathbb{R}_{>0})$ . Let $t\in\mathbb{R}$ . Let $Z,N,\epsilon\in\mathbb{R}_{>0}$ with $N\leq Z^{3}$ and $\epsilon\leq 1$ . Then for some real $A_{6}=A_{6}(F,\epsilon)>0$ , we have

(7.10)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\,\Bigl{\lvert}{\sum_{n\in\mathcal{N}^{\bm{c}}}D(n/N)\cdot n^{-it}S^{\natural}_{\bm{c}}(n)}\Bigr{\rvert}^{2-\epsilon}\ll_{F,D,\epsilon}(1+\lvert t\rvert)^{A_{6}}Z^{m}N^{(2-\epsilon)/2}.

Proposition 7.5.

Assume Conjectures 1.2 and 6.9. Then Conjecture 7.4 holds.

Proof.

Let $\nu_{2}\in C^{\infty}_{c}(\mathbb{R}_{>0})$ be as in §5, so that $\operatorname{Supp}{\nu_{2}}\subseteq[1,2]$ and we have (5.4). In view of (7.1) and the factorization $\Phi^{\operatorname{G}}=\Phi^{\bm{c},1}\Phi^{\bm{c},2}\Phi^{\bm{c},3}$ , we may use Lemma 5.2 (with $k=3$ and $a(\bm{n})=\prod_{1\leq j\leq 3}(n_{j}^{-it}a_{\bm{c},j}(n_{j}))$ , and $f(\bm{r})=D(r_{1}r_{2}r_{3}/N)$ ) to write

(7.11)

\sum_{n\in\mathcal{N}^{\bm{c}}}D(n/N)\cdot n^{-it}S^{\natural}_{\bm{c}}(n)=(2\pi)^{-3}\int_{\bm{N}\geq 1/2}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{3}}d\bm{t}\,g_{\bm{N}}^{\vee}(i\bm{t})\prod_{1\leq j\leq 3}\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t}),

where $\bm{N}=(N_{1},N_{2},N_{3})$ , where $g_{\bm{N}}(\bm{r})\colonequals D(r_{1}r_{2}r_{3}/N)\prod_{1\leq j\leq 3}\nu_{2}(r_{j}/N_{j})$ , and where

(7.12)

\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t})=\Sigma_{10,\bm{N}}^{\bm{c},j}(t_{1},t_{2},t_{3})\colonequals\sum_{n_{j}\geq 1}\nu_{2}(n_{j}/N_{j})n_{j}^{-i(t_{j}+t)}a_{\bm{c},j}(n_{j}).

(Note that $N$ , $t$ are independent of $\bm{N}$ , $\bm{t}$ .) For all $\bm{N}\geq 1/2$ and $b\geq 0$ , Proposition 5.1 gives

(7.13)

g_{\bm{N}}^{\vee}(i\bm{t})\ll_{b,D,\nu_{2}}(1+\lVert\bm{t}\rVert)^{-b}.

Fix an integer $A\geq 1$ for which $\operatorname{Supp}{D}\subseteq[A^{-1},A]$ . If $\bm{r}\in\operatorname{Supp}{g_{\bm{N}}}$ , then $N/A\leq r_{1}r_{2}r_{3}\leq AN$ and $N_{j}\leq r_{j}\leq 2N_{j}$ for all $j$ , so $\bm{N}$ lies in the set

(7.14)

\mathscr{R}_{10}\colonequals\{\bm{N}\geq 1/2:N_{1}N_{2}N_{3}\in[N/8A,AN]\}.

Thus the equality (7.11) remains true if we restrict the integral over $\bm{N}\geq 1/2$ to the region $\mathscr{R}_{10}$ . Now set $W_{1}(\bm{N},\bm{t})\colonequals N_{1}^{\eta}\cdot\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert$ and $W_{2}(\bm{N},\bm{t})\colonequals N_{1}^{-(\beta-1)\eta}\cdot\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert$ , for a small constant $\eta>0$ to be chosen later. Let $\mathcal{I}_{1}\colonequals\int_{\mathscr{R}_{10}\times\mathbb{R}^{3}}d^{\times}\bm{N}\,d\bm{t}\,W_{1}(\bm{N},\bm{t})$ . Letting $\beta\colonequals 2-\epsilon\in[1,2)$ , and using Hölder over $(\log\bm{N},\bm{t})\subseteq\mathbb{R}^{3}\times\mathbb{R}^{3}$ (restricted to $\bm{N}\in\mathscr{R}_{10}$ ), we obtain

(7.15)

\Bigl{\lvert}{\sum_{n\in\mathcal{N}^{\bm{c}}}D(n/N)n^{-it}S^{\natural}_{\bm{c}}(n)}\Bigr{\rvert}^{\beta}\leq\mathcal{I}_{1}^{\beta-1}\cdot\int_{\mathscr{R}_{10}\times\mathbb{R}^{3}}d^{\times}\bm{N}\,d\bm{t}\,W_{2}(\bm{N},\bm{t})\prod_{1\leq j\leq 3}\lvert\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t})\rvert^{\beta},

since $\beta\geq 1$ and $W_{1}(\bm{N},\bm{t})^{\beta-1}\cdot W_{2}(\bm{N},\bm{t})=\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert^{\beta}$ .

By (7.13) and (7.14), we have $\mathcal{I}_{1}\ll_{D,\nu_{2}}\int_{\mathscr{R}_{10}}d^{\times}\bm{N}\,N_{1}^{\eta}\ll_{A,\eta}N^{\eta}\int_{N_{2},N_{3}\geq 1/2}\frac{d^{\times}{N_{2}}\,d^{\times}{N_{3}}}{(N_{2}N_{3})^{\eta}}\ll_{\eta}N^{\eta}$ . Upon summing (7.15) over $\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}$ , we thus find that the left-hand side of (7.10) is

(7.16)

\ll_{D,\nu_{2},\eta}\int_{\mathscr{R}_{10}\times\mathbb{R}^{3}}d^{\times}\bm{N}\,d\bm{t}\,(N/N_{1})^{(\beta-1)\eta}\cdot\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\prod_{1\leq j\leq 3}\lvert\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t})\rvert^{\beta}.

Now let $(\gamma_{1},\gamma_{2},\gamma_{3})\colonequals(2,4\beta/\epsilon,4\beta/\epsilon)$ . Then $\sum_{1\leq j\leq 3}\beta/\gamma_{j}=1$ (since $\beta=2-\epsilon$ ), so by Hölder over $\bm{c}$ (writing $\heartsuit_{10}=\prod_{1\leq j\leq 3}\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t})$ and $\mathscr{M}_{j}=\lVert\Sigma_{10,\bm{N}}^{\bm{c},j}(\bm{t})\rVert_{\ell^{\gamma_{j}}_{\bm{c}}(\mathcal{S}_{1}\cap[-Z,Z]^{m})}^{\gamma_{j}}$ for brevity),

(7.17)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\lvert\heartsuit_{10}\rvert^{\beta}=\lVert\heartsuit_{10}\rVert_{\ell^{\beta}_{\bm{c}}(\mathcal{S}_{1}\cap[-Z,Z]^{m})}^{\beta}\leq\prod_{1\leq j\leq 3}\mathscr{M}_{j}^{\beta/\gamma_{j}}.

We now bound the necessary $\ell^{\gamma_{j}}$ -norms. First, $\mathscr{M}_{1}\ll_{\nu_{2}}(1+\lvert t_{1}+t\rvert)^{A_{3}}(Z+N_{1}^{1/3})^{m}N_{1}^{\gamma_{1}/2}$ , by (7.12) and Conjecture 6.9 (with $Z+N_{1}^{1/3}$ , $N_{1}$ in place of $Z$ , $N$ ). Second, by (7.12), (7.5), and (6.37) (with $A=\gamma_{2}$ , with $f=\nu_{2}$ , and with $Z+N_{2}^{1/3}$ , $N_{2}^{1/2}$ in place of $Z$ , $N$ ), we have $\mathscr{M}_{2}\ll_{\gamma_{2},\nu_{2}}(1+\lvert t_{2}+t\rvert)^{\gamma_{2}}(Z+N_{2}^{1/3})^{m}(N_{2}^{1/2})^{(1-\eta_{3}(\gamma_{2}))\gamma_{2}}$ . Third, by (7.12), the bound $\nu_{2}(n_{3}/N_{3})n_{3}^{-i(t_{3}+t)}\ll_{\nu_{2}}1$ , and Corollary 7.3, we have $\mathscr{M}_{3}\ll_{\gamma_{3},\nu_{2}}Z^{m}N_{3}^{11\gamma_{3}/30}$ .

Plugging into (7.17), and writing $1+\lvert t_{j}+t\rvert\leq(1+\lVert\bm{t}\rVert)(1+\lvert t\rvert)$ , we get that if $\bm{N}\in\mathscr{R}_{10}$ , then the left-hand side of (7.17) is $\ll_{D,\nu_{2},\beta}(1+\lVert\bm{t}\rVert)^{A_{6}}(1+\lvert t\rvert)^{A_{6}}$ times

Z^{\sum_{j}m\beta/\gamma_{j}}N_{1}^{\beta/2}N_{2}^{(1-\eta_{3}(\gamma_{2}))\beta/2}N_{3}^{11\beta/30}\asymp_{D,\beta}Z^{m}(N^{1/2}N_{2}^{-\eta_{3}(\gamma_{2})/2}N_{3}^{-4/30})^{\beta},

where $A_{6}=(A_{3}\beta/\gamma_{1}+\gamma_{2}\beta/\gamma_{2})=(A_{3}/2+1)\beta$ . Upon plugging this result into (7.16), we get that the left-hand side of (7.10) is at most $O_{D,\nu_{2},\beta}(1)$ times

(7.18)

\int_{\mathscr{R}_{10}\times\mathbb{R}^{3}}d^{\times}\bm{N}\,d\bm{t}\,(N/N_{1})^{(\beta-1)\eta}\cdot\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert\cdot(1+\lVert\bm{t}\rVert)^{A_{6}}\frac{(1+\lvert t\rvert)^{A_{6}}Z^{m}N^{\beta/2}}{(N_{2}^{\eta_{3}(\gamma_{2})/2}N_{3}^{4/30})^{\beta}}.

Finally, let $\eta=\min(\eta_{3}(\gamma_{2})/2,4/30)$ . Then for each $\bm{N}\in\mathscr{R}_{10}$ , we have $(N/N_{1})^{\eta}\ll_{D,\beta}N_{2}^{\eta_{3}(\gamma_{2})/2}N_{3}^{4/30}$ . Since $\beta-1\geq 0$ , it follows that the quantity (7.18) is

\begin{split}\ll_{D,\beta}\int_{\mathscr{R}_{10}\times\mathbb{R}^{3}}d^{\times}\bm{N}\,d\bm{t}\,\lvert g_{\bm{N}}^{\vee}(i\bm{t})\rvert\cdot(1+\lVert\bm{t}\rVert)^{A_{6}}\frac{(1+\lvert t\rvert)^{A_{6}}Z^{m}N^{\beta/2}}{N_{2}^{\eta_{3}(\gamma_{2})/2}N_{3}^{4/30}}\ll_{D,\beta}(1+\lvert t\rvert)^{A_{6}}Z^{m}N^{\beta/2},\end{split}

where we have used (7.13) to integrate over $\bm{t}$ , and (7.14) to integrate first over $N_{1}$ (given $N_{2}$ , $N_{3}$ ) and then over $N_{2},N_{3}\geq 1/2$ . Therefore, (7.10) holds (with $A_{6}=(A_{3}/2+1)(2-\epsilon)$ ). ∎

7.3. Handling variation of “error factors” for small fixed “error moduli”

We would like to build on Propositions 6.13 and 6.14, but we must first improve our understanding of certain local factors. For each $\bm{c}\in\mathcal{S}_{1}$ , recall $\Phi^{\bm{c},j}(s)$ , $a_{\bm{c},j}(n)$ from Definition 7.1, and let $a^{\prime}_{\bm{c}}(n)$ denote the $n$ th coefficient of the Dirichlet series

(7.19)

\Phi^{\bm{c},2}\Phi^{\bm{c},3}=\Phi^{\operatorname{G}}/\Phi^{\bm{c},1}=\Phi^{\operatorname{G}}(\bm{c},s)L(s,V_{\bm{c}})L(1/2+s,V)\zeta(2s),

so that for all $n\geq 1$ , we have

(7.20)

S^{\natural}_{\bm{c}}(n)\bm{1}_{n\in\mathcal{N}^{\bm{c}}}=\sum_{n_{1}d=n}a_{\bm{c},1}(n_{1})a^{\prime}_{\bm{c}}(d).

By Proposition 7.2, $a_{\bm{c},3}(n)\ll_{\epsilon}n^{\epsilon}$ , and by (3.2) and Definition 7.1, $a_{\bm{c},2}(n)\ll_{\epsilon}n^{\epsilon}$ ; so certainly

(7.21)

a^{\prime}_{\bm{c}}(d)\ll_{\epsilon}d^{\epsilon}.

Moreover, if $d$ is small (or fixed), we would like $a^{\prime}_{\bm{c}}(d)$ to not vary too wildly with $\bm{c}$ .

Given $n\in\mathbb{Z}_{\geq 1}$ , what data does $a^{\prime}_{\bm{c}}(n)$ depend on? Note that $L(1/2+s,V)\zeta(2s)$ is fixed (in terms of $F$ ). So by (7.19) and (7.1), the coefficient $a^{\prime}_{\bm{c}}(n)$ is determined by the residue class $\bm{c}+n\mathbb{Z}^{m}$ and the local factors $L_{p}(s,V_{\bm{c}})$ for $p\mid n$ . Therefore, if we define $l(p,\bm{c})$ as in Lemma 4.4, then $a^{\prime}_{\bm{c}}(n)$ is determined by the residue class $\bm{c}+r(n,\bm{c})\mathbb{Z}^{m}$ , where

(7.22)

r(n,\bm{c})\colonequals\prod_{p\mid n}\operatorname{lcm}(p^{v_{p}(n)},p^{l(p,\bm{c})+1}).

Proposition 7.6.

Let $n,q\geq 1$ . Let $\bm{a},\bm{b}\in\mathcal{S}_{1}$ and suppose $\bm{a}\equiv\bm{b}\bmod{q}$ . Let $p\mid n$ be a prime. Then $v_{p}(r(n,\bm{a}))\leq v_{p}(q)$ if and only if $v_{p}(r(n,\bm{b}))\leq v_{p}(q)$ .

Proof.

By symmetry, it suffices to prove the “if” direction. So, say $v_{p}(q)\geq v_{p}(r(n,\bm{b}))$ . Then $v_{p}(q)\geq l(p,\bm{b})+1$ , so $l(p,\bm{a})=l(p,\bm{b})$ by (4.6). Thus $v_{p}(r(n,\bm{a}))=v_{p}(r(n,\bm{b}))\leq v_{p}(q)$ . ∎

In what follows, recall the notation $q_{\mathcal{R}}$ from Definition 6.11.

Definition 7.7.

For every $n\geq 1$ , let $\mathscr{E}(n)$ be the set of residue classes $\mathcal{R}\subseteq\mathbb{Z}^{m}$ for which there exists a tuple $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ with $r(n,\bm{c})\nmid q_{\mathcal{R}}$ .

We now construct a partition $\mathscr{P}(n,k)=\{\mathcal{R}\}$ of $\mathbb{Z}^{m}$ into residue classes $\mathcal{R}\subseteq\mathbb{Z}^{m}$ .

Definition 7.8.

Fix integers $n,k\geq 1$ . We define $\mathscr{P}(n,k)$ through a recursive decomposition process. Let $\mathscr{S}_{0}\colonequals\{\bm{a}+n\mathbb{Z}^{m}:1\leq\bm{a}\leq n\}$ denote the partition of $\mathbb{Z}^{m}$ into the $n^{m}$ residue classes modulo $n$ . For each $j\geq 0$ , define $\mathscr{S}_{j+1}$ in terms of $\mathscr{S}_{j}$ as follows:

(1)

If possible, choose a residue class $\mathcal{R}\in\mathscr{S}_{j}\cap\mathscr{E}(n)$ with $q_{\mathcal{R}}\leq n^{k-1}$ . Otherwise, let $\mathscr{S}_{j+1}\colonequals\mathscr{S}_{j}$ , and skip step (2).

(2)

Write $\mathcal{R}=\bm{a}+q\mathbb{Z}^{m}$ , with $q\geq 1$ . Choose $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ with $r(n,\bm{c})\nmid q$ . Choose a prime $p\mid n$ with $v_{p}(r(n,\bm{c}))>v_{p}(q)$ . Create $\mathscr{S}_{j+1}$ by replacing the element $\mathcal{R}\in\mathscr{S}_{j}$ with the $p^{m}$ lifted residue classes $(\bm{a}+q\bm{i})+pq\mathbb{Z}^{m}$ (with $1\leq\bm{i}\leq p$ , say). Formally,

\mathscr{S}_{j+1}\colonequals(\mathscr{S}_{j}\setminus\{\mathcal{R}\})\cup\{(\bm{a}+q\bm{i})+pq\mathbb{Z}^{m}:1\leq\bm{i}\leq p\}.

Step (2) can only occur finitely many times (because we require $q_{\mathcal{R}}\leq n^{k-1}$ in step (1)). Let $j_{0}\colonequals\min{\{j\geq 0:\mathscr{S}_{j+1}=\mathscr{S}_{j}\}}$ . Let $\mathscr{P}(n,k)\colonequals\mathscr{S}_{j_{0}}$ .⁵⁵5The result may depend on the choices we make at each step, but all of our estimates based on $\mathscr{P}(n,k)$ will apply uniformly over all possible outcomes. Let $\mathscr{E}(n,k)\colonequals\mathscr{P}(n,k)\cap\mathscr{E}(n)$ .

In Definition 7.8, we allow the initial $\mathscr{S}_{0}$ to branch into many different moduli. If we did not do this, then to control $a^{\prime}_{\bm{c}}(n)$ for $n\leq M$ might require us to work with moduli exponentially large in $M$ (for some values of $n$ ), which would be fatal to our approach to Theorem 1.9 (though perhaps OK for Theorem 1.6). We will eventually apply Propositions 6.13 and 6.14 in residue classes $\mathcal{R}\in\mathscr{P}(n,k)\setminus\mathscr{E}(n)$ , for some values of $n,k\geq 1$ . We first unravel the structure of $\mathscr{P}(n,k)$ , and provide some control on the exceptional set $\mathscr{E}(n)$ .

Lemma 7.9.

Let $n,k\geq 1$ . Suppose $\mathcal{R}\in\mathscr{S}_{j}$ for some $j\geq 0$ . Suppose $q_{\mathcal{R}}>n$ . Then $j\geq 1$ . Furthermore, there exist an index $i\in[0,j-1]$ , a prime $p\mid n$ , and a residue class $\mathcal{R}^{\prime}\in\mathscr{S}_{i}\cap\mathscr{E}(n)$ of modulus $q_{\mathcal{R}^{\prime}}\leq n^{k-1}$ with $\mathcal{R}\subseteq\mathcal{R}^{\prime}$ and $q_{\mathcal{R}}/q_{\mathcal{R}^{\prime}}=p$ , such that $v_{p}(q_{\mathcal{R}})\leq v_{p}(r(n,\bm{c}))$ holds for all $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}^{\prime}$ .

Proof.

Each element of $\mathscr{S}_{0}$ has modulus $n$ , so $\mathcal{R}\notin\mathscr{S}_{0}$ ; in particular, $j\neq 0$ , so $j\geq 1$ . Choose $i\in[0,j-1]$ maximal with $\mathcal{R}\notin\mathscr{S}_{i}$ . Then by Definition 7.8, there exists a residue class $\mathcal{R}^{\prime}\in\mathscr{S}_{i}\cap\mathscr{E}(n)$ , a tuple $\bm{c}^{\prime}\in\mathcal{S}_{1}\cap\mathcal{R}^{\prime}$ , and a prime $p\mid n$ , with $q_{\mathcal{R}^{\prime}}\leq n^{k-1}$ and $v_{p}(r(n,\bm{c}^{\prime}))>v_{p}(q_{\mathcal{R}^{\prime}})$ , such that $\mathcal{R}\subseteq\mathcal{R}^{\prime}$ and $q_{\mathcal{R}}/q_{\mathcal{R}^{\prime}}=p$ . Now let $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}^{\prime}$ . By Proposition 7.6, $v_{p}(r(n,\bm{c}))>v_{p}(q_{\mathcal{R}^{\prime}})$ (since $v_{p}(r(n,\bm{c}^{\prime}))>v_{p}(q_{\mathcal{R}^{\prime}})$ ). So $v_{p}(r(n,\bm{c}))\geq 1+v_{p}(q_{\mathcal{R}^{\prime}})=v_{p}(q_{\mathcal{R}})$ . ∎

Proposition 7.10.

Let $n,k\geq 1$ . Let $\mathcal{R}\in\mathscr{P}(n,k)$ . Then $q_{\mathcal{R}}\leq n^{k}$ . Furthermore, if $n\geq 2$ and $\mathcal{R}\in\mathscr{P}(n,k)\setminus\mathscr{E}(n)$ , then $\mathcal{R}\subseteq\mathcal{S}_{1}$ .

Proof.

By Lemma 7.9, we either have $q_{\mathcal{R}}\leq n$ , or $q_{\mathcal{R}}=pq_{\mathcal{R}^{\prime}}$ with $p\mid n$ and $q_{\mathcal{R}^{\prime}}\leq n^{k-1}$ . So $q_{\mathcal{R}}\leq n^{k}$ . Now suppose $n\geq 2$ and $\mathcal{R}\in\mathscr{P}(n,k)\setminus\mathscr{E}(n)$ . Choose $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ (possible by (2.4)). Then $r(n,\bm{c})\mid q_{\mathcal{R}}$ , by the definition of $\mathscr{E}(n)$ . Now choose $p\mid n$ (possible since $n\geq 2$ ). Then $p^{l(p,\bm{c})+1}\mid q_{\mathcal{R}}$ , by (7.22). But by the definition of $l(p,\bm{c})$ , we have $\Delta(\bm{c}^{\prime})\neq 0$ for all $\bm{c}^{\prime}\equiv\bm{c}\bmod{p^{l(p,\bm{c})+1}}$ . Thus $\mathcal{R}\subseteq\mathcal{S}_{1}$ (since $\bm{c}\in\mathcal{R}$ ). ∎

Proposition 7.11.

Let $n,k\geq 2$ . If $\mathcal{R}\in\mathscr{E}(n,k)$ , then $r(n,\bm{c})\geq n^{k-2}$ for all $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ .

Proof.

Suppose $\mathcal{R}\in\mathscr{E}(n,k)$ . By Definition 7.8, $\mathscr{E}(n,k)=\mathscr{S}_{j_{0}}\cap\mathscr{E}(n)$ . So $\mathcal{R}\in\mathscr{S}_{j_{0}}\cap\mathscr{E}(n)$ , whence $q_{\mathcal{R}}>n^{k-1}$ (or else we would have $\mathscr{S}_{j_{0}+1}\neq\mathscr{S}_{j_{0}}$ by the algorithm in Definition 7.8). By Lemma 7.9 (applied repeatedly), there exists a sequence of primes $p_{1},\dots,p_{s}\mid n$ , with $p_{1}\cdots p_{s}\mid q_{\mathcal{R}}$ and $q_{\mathcal{R}}/(p_{1}\cdots p_{s-1})>n\geq q_{\mathcal{R}}/(p_{1}\cdots p_{s})$ , such that

(7.23)

v_{p_{i}}(q_{\mathcal{R}}/(p_{1}\cdots p_{i-1}))\leq v_{p_{i}}(r(n,\bm{c}))

holds for all $i\in\{1,2,\dots,s\}$ and $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ . If for each $p\mid p_{1}\cdots p_{s}$ , we apply (7.23) with $i=\min\{1\leq u\leq s:p_{u}=p\}$ , then we get $v_{p}(q_{\mathcal{R}})\leq v_{p}(r(n,\bm{c}))$ . Since $p_{1}\cdots p_{s}\mid q_{\mathcal{R}}$ , it follows that $p_{1}\cdots p_{s}\mid r(n,\bm{c})$ , and thus $r(n,\bm{c})\geq p_{1}\cdots p_{s}\geq q_{\mathcal{R}}/n>n^{k-2}$ . ∎

Let $\mu(\mathcal{R})\colonequals q_{\mathcal{R}}^{-m}$ be the density of a residue class $\mathcal{R}$ in $\mathbb{Z}^{m}$ .

Lemma 7.12 (KL’).

Let $n\geq 2$ . Then $\lim_{k\to\infty}\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mu(\mathcal{R})=0$ .

Proof.

Suppose $\mathcal{R}\in\mathscr{E}(n,k)$ and $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ , where $k\geq 3$ . Then by Proposition 7.11, we have $r(n,\bm{c})\geq n^{k-2}$ . But by (7.22), we have $r(n,\bm{c})\mid\prod_{p\mid n}p^{v_{p}(n)+l(p,\bm{c})}=n\prod_{p\mid n}p^{l(p,\bm{c})}$ . Therefore (since $n\geq 2$ ), there exists a prime $p\mid n$ with $l(p,\bm{c})\geq k-3$ . It follows that

\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mathbb{E}_{\bm{c}\in[-Z,Z]^{m}}[\bm{1}_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}}]\leq\sum_{p\mid n}\mathbb{E}_{\bm{c}\in[-Z,Z]^{m}}[\bm{1}_{\bm{c}\in\mathcal{S}_{1}}\bm{1}_{l(p,\bm{c})\geq k-3}]

for all reals $Z\geq 1$ . Taking $Z\to\infty$ (using (2.4) on the left-hand side, and Lemma 4.4 on the right-hand side; cf. the proof of Proposition 6.1), we get

(7.24)

\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mu(\mathcal{R})\leq\sum_{p\mid n}\mathbb{E}_{\bm{c}\in\mathbb{Z}_{p}^{m}}[\bm{1}_{\Delta(\bm{c})\neq 0}\bm{1}_{l(p,\bm{c})\geq k-3}].

But by Lemma 4.4, the right-hand side of (7.24) tends to $0$ as $k\to\infty$ . ∎

Lemma 7.13 (EKL’).

Assume Conjecture 1.11 (EKL). Let $n\geq 2$ and $k\geq 3$ . Then $\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mu(\mathcal{R})\ll_{H,\epsilon}n^{k\epsilon}\cdot n^{-(k-3)/\deg{H}}$ (uniformly over $n$ and $k$ ).

Proof.

Suppose $\mathcal{R}\in\mathscr{E}(n,k)$ and $\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}$ . As in the proof of Lemma 7.12, we have $r(n,\bm{c})\geq n^{k-2}$ and $r(n,\bm{c})\mid n\prod_{p\mid n}p^{l(p,\bm{c})}$ . By Remark 4.5, we have $\prod_{p\mid n}p^{l(p,\bm{c})}\mid H(\bm{c})$ . But $n\mid r(n,\bm{c})\mid n^{\infty}$ by (7.22). Hence $H(\bm{c})$ is divisible by the integer $q=r(n,\bm{c})/n\geq n^{k-3}$ , where $q\mid n^{\infty}$ . Since every prime factor of $q$ is $\leq n$ , there exists $u\mid q$ with $n^{k-3}\leq u<n^{k-2}$ . On the other hand, if $\bm{c}\in\mathcal{S}_{0}\cap\mathcal{R}$ , then $H(\bm{c})=0$ (since $\Delta\mid H$ ), so $n^{k-3}\mid H(\bm{c})$ . So for every $\bm{c}\in\mathcal{R}$ (if $\mathcal{R}\in\mathscr{E}(n,k)$ ), there exists $u\in[n^{k-3},n^{k-2})$ , with $u\mid n^{\infty}$ , such that $u\mid H(\bm{c})$ . Thus

\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mathbb{E}_{\bm{c}\in[-Z,Z]^{m}}[\bm{1}_{\bm{c}\in\mathcal{R}}]\leq\sum_{n^{k-3}\leq u<n^{k-2}:\,u\mid n^{\infty}}\mathbb{E}_{\bm{c}\in[-Z,Z]^{m}}[\bm{1}_{u\mid H(\bm{c})}]

for all reals $Z\geq 1$ . Taking $Z\to\infty$ (using (4.3) on the right-hand side), we get

\sum_{\mathcal{R}\in\mathscr{E}(n,k)}\mu(\mathcal{R})\ll_{H}\sum_{n^{k-3}\leq u<n^{k-2}:\,u\mid n^{\infty}}u^{-1/\deg{H}}\leq\sum_{u\geq n^{k-3}:\,u\mid n^{\infty}}u^{-1/\deg{H}}.

But $\sum_{u\geq N:\,u\mid n^{\infty}}u^{-\beta}\leq N^{\epsilon-\beta}\sum_{u\geq 1:\,u\mid n^{\infty}}u^{-\epsilon}\ll_{\epsilon}N^{\epsilon-\beta}n^{\epsilon}$ , for all $N,\beta,\epsilon\in\mathbb{R}_{>0}$ with $\epsilon<\beta$ . ∎

7.4. From (RA1’E) to (RA1’E’)

We now build on Propositions 6.13 and 6.14. Let $I\subseteq\mathbb{R}_{>0}$ be a compact set. Let $\nu=\nu_{\bm{c}}(r)$ be a smooth function $\mathbb{R}^{m}\times\mathbb{R}\to\mathbb{C},\,(\bm{c},r)\mapsto\nu_{\bm{c}}(r)$ , supported on $[-1,1]^{m}\times I$ . Given $(\bm{a},n_{0})\in\mathbb{Z}^{m}\times\mathbb{Z}_{\geq 1}$ and reals $Z,N\geq 1$ , let

(7.25)

\Sigma_{11}^{\bm{a},n_{0}}(\nu,Z,N)\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}\sum_{n\geq 1}\nu_{\bm{c}/Z}(n/N)S^{\natural}_{\bm{c}}(n)\bm{1}_{n\in\mathcal{N}^{\bm{c}}}

Conjecture 7.14 (RA1 $o$ ’E’).

Let $M\in\mathbb{R}_{\geq 1}$ , and let $n_{0}\leq M^{9/10}$ be a positive integer. Let $Z,N\geq 2M$ be reals with $N\leq Z^{3}$ . Then $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}(\nu,Z,N)\rvert\ll_{F,I}Z^{m}N^{1/2}\cdot(o_{F;M\to\infty}(1)+o_{F,M;Z\to\infty}(1))\cdot\mathcal{M}_{1,A_{7}}$ for some $A_{7}=A_{7}(F)>0$ , where $\mathcal{M}_{1,k}$ is defined as in (6.24).

The intermediate parameter $M$ here may seem strange, but it will ease our exposition.

Proposition 7.15.

Assume Conjectures 1.2, 6.9, and 6.12. Then Conjecture 7.14 holds.

Proof.

Plugging (7.20) into (7.25) reveals the equality

\Sigma_{11}^{\bm{a},n_{0}}(\nu,Z,N)=\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}\sum_{n_{1},d\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1})a^{\prime}_{\bm{c}}(d).

Fix a function $\upsilon_{0}\in C^{\infty}_{c}(\mathbb{R})$ with $\operatorname{Supp}{\upsilon_{0}}\subseteq[-1,1]$ and $\upsilon_{0}|_{[-1/2,1/2]}=1$ . Let $L\geq 1$ denote a real number to be chosen later. We first analyze the piece

\Sigma_{12}^{\bm{a},n_{0}}\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}\sum_{n_{1},d\geq 1}(1-\upsilon_{0}(d/L))\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1})a^{\prime}_{\bm{c}}(d)

of $\Sigma_{11}^{\bm{a},n_{0}}$ . For later reference, note that (since $\operatorname{Supp}{\upsilon_{0}}\subseteq[-1,1]$ )

(7.26)		$\displaystyle\Sigma_{11}^{\bm{a},n_{0}}-\Sigma_{12}^{\bm{a},n_{0}}$	$\displaystyle=\sum_{d\leq L}\upsilon_{0}(d/L)\Sigma_{13}^{\bm{a},n_{0}}(d),$
(7.27)		$\displaystyle\textnormal{where}\quad\Sigma_{13}^{\bm{a},n_{0}}(d)$	$\displaystyle\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}a^{\prime}_{\bm{c}}(d)\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1}).$

We bound $\Sigma_{12}^{\bm{a},n_{0}}$ using the Hölder technique behind (the simplest case, $\epsilon=1$ , of) Proposition 7.5. Since $a^{\prime}_{\bm{c}}(d)$ is the $d$ th coefficient of the Dirichlet series $\Phi^{\bm{c},2}(s)\Phi^{\bm{c},3}(s)$ (see §7.3), we may write $a^{\prime}_{\bm{c}}$ in terms of $a_{\bm{c},2}$ , $a_{\bm{c},3}$ , and then apply Lemma 5.2 (with $k=3$ and $a(\bm{n})=\prod_{1\leq j\leq 3}a_{\bm{c},j}(n_{j})$ , and $f(\bm{r})=(1-\upsilon_{0}(r_{2}r_{3}/L))\nu_{\bm{c}/Z}(r_{1}r_{2}r_{3}/N)$ ), to get

(7.28)

\Sigma_{12}^{\bm{a},n_{0}}=\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}(2\pi)^{-3}\int_{\bm{N}\in[1/2,\infty)^{3}}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{3}}d\bm{t}\,g_{\bm{c},\bm{N}}^{\vee}(i\bm{t})\prod_{1\leq j\leq 3}\Sigma_{12,\bm{N}}^{\bm{c},j}(\bm{t})

(cf. (7.11)), where $g_{\bm{c},\bm{N}}(\bm{r})\colonequals(1-\upsilon_{0}(r_{2}r_{3}/L))\nu_{\bm{c}/Z}(r_{1}r_{2}r_{3}/N)\prod_{1\leq j\leq 3}\nu_{2}(r_{j}/N_{j})$ and

\Sigma_{12,\bm{N}}^{\bm{c},j}(\bm{t})\colonequals\bm{1}_{\bm{c}\in[-Z,Z]^{m}}\sum_{n_{j}\geq 1}\nu_{2}(n_{j}/N_{j})n_{j}^{-it_{j}}a_{\bm{c},j}(n_{j}).

For every integer $b\geq 0$ , Proposition 5.1 and (6.24) imply (uniformly over $\bm{c}$ , $\bm{N}$ , $\bm{t}$ )

(7.29)

g_{\bm{c},\bm{N}}^{\vee}(i\bm{t})\ll_{b}\mathcal{M}_{1,b}(\nu)(1+\lVert\bm{t}\rVert)^{-b}

(where the implied constant may depend on $\upsilon_{0}$ , $\nu_{2}$ as well as $b$ ).

Since $1-\upsilon_{0}$ , $\nu_{\bm{c}/Z}$ , $\nu_{2}$ are supported on $\mathbb{R}\setminus[-1/2,1/2]$ , $I$ , $[1,2]$ , respectively, we have $g_{\bm{c},\bm{N}}(\bm{r})=0$ unless $r_{2}r_{3}\geq L/2$ , $r_{1}r_{2}r_{3}\in N\cdot I$ , and $N_{j}\leq r_{j}\leq 2N_{j}$ for all $j$ . Fix an integer $A\geq 1$ satisfying $I\subseteq[A^{-1},A]$ . Then $g_{\bm{c},\bm{N}}=0$ identically unless $\bm{N}$ lies in the set

(7.30)

\mathscr{R}_{12}\colonequals\{\bm{N}\geq 1/2:N_{2}N_{3}\geq L/8,\;N_{1}N_{2}N_{3}\in[N/8A,AN]\}

(cf. the region $\mathscr{R}_{10}$ from (7.14)). So (7.28) holds even if we restrict $\bm{N}$ to $\mathscr{R}_{12}$ .

But if $\bm{N}\in\mathscr{R}_{12}$ and $\bm{t}\in\mathbb{R}^{3}$ , then (7.17) (with $\beta=1$ , $(\gamma_{1},\gamma_{2},\gamma_{3})=(2,4,4)$ , $t=0$ ) and the subsequent arguments up to (7.18) furnish (via Conjectures 1.2 and 6.9) the bound

(7.31)

\int_{\bm{N}\in\mathscr{R}_{12}}d^{\times}\bm{N}\sum_{\bm{c}\in\mathcal{S}_{1}}\prod_{1\leq j\leq 3}\lvert\Sigma_{12,\bm{N}}^{\bm{c},j}(\bm{t})\rvert\ll_{I}\int_{\bm{N}\in\mathscr{R}_{12}}d^{\times}\bm{N}\,(1+\lVert\bm{t}\rVert)^{A_{6}}\frac{Z^{m}N^{1/2}}{N_{2}^{\eta_{3}(4)/2}N_{3}^{4/30}}

(where $A_{6}=(A_{3}/2+1)\beta=A_{3}/2+1$ , and $\eta_{3}(4)$ is as in Proposition 6.18). Upon taking absolute values in (7.28) (after restricting $\bm{N}$ to $\mathscr{R}_{12}$ ), summing over $1\leq\bm{a}\leq n_{0}$ , plugging in (7.29) (with $b=\lceil A_{6}+4\rceil$ ) and then (7.31), and integrating over $\bm{t}\in\mathbb{R}^{3}$ , we conclude that

(7.32)

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{12}^{\bm{a},n_{0}}\rvert\ll_{I}\mathcal{M}_{1,b}(\nu)\int_{\bm{N}\in\mathscr{R}_{12}}d^{\times}\bm{N}\,\frac{Z^{m}N^{1/2}}{N_{2}^{\eta_{3}(4)/2}N_{3}^{4/30}}\ll_{I}\mathcal{M}_{1,b}(\nu)\frac{Z^{m}N^{1/2}}{L^{\eta_{4}}},

where $b=\lceil A_{6}+4\rceil$ and $\eta_{4}=0.9\min(\eta_{3}(4)/2,4/30)$ .

We now turn to the sums $\Sigma_{13}^{\bm{a},n_{0}}(d)$ , for $d\leq L$ . We first treat $d=1$ . By (7.27), we have $\Sigma_{13}^{\bm{a},n_{0}}(1)=\sum_{\bm{c}\in\mathcal{S}_{1}:\,\bm{c}\equiv\bm{a}\bmod{n_{0}}}\sum_{n\geq 1}\nu_{\bm{c}/Z}(n/N)a_{\bm{c},1}(n)$ . Therefore, by the triangle inequality, $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13}^{\bm{a},n_{0}}(1)\rvert$ is at most the sum of the quantity (6.25) (with $\mathscr{P}=\{\bm{a}+n_{0}\mathbb{Z}^{m}:1\leq\bm{a}\leq n_{0}\}$ ) and the left-hand side of (6.33) (with $\mathcal{S}=\mathcal{S}_{1}$ ). So by Conjecture 6.12 (applicable since $Z,N\geq 2M\geq 2n_{0}=2Q(\mathscr{P})$ ) and Lemma 6.15 (with $\theta=m$ and $Q(\mathscr{P})=n_{0}\leq M$ ), the sum $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13}^{\bm{a},n_{0}}(1)\rvert$ is at most $O_{I,\epsilon}(\mathfrak{B}_{0}(N,\epsilon))$ , where $\mathfrak{B}_{0}(N,\epsilon)$ denotes the the expression⁶⁶6The replacement of $n_{0}$ with the weaker (larger) $M$ here may seem strange, but it will be convenient later.

(7.33)

Z^{m}N^{1/2}o_{M;Z\to\infty}(1)\mathcal{M}_{1,A_{4}}+Z^{m}N^{1/3+\epsilon}M^{\epsilon}\mathcal{M}_{1,0}.

If $\epsilon$ is sufficiently small, then (since $M\leq N$ )

(7.34)

\mathfrak{B}_{0}(N,\epsilon)\leq Z^{m}N^{1/2}(o_{M;Z\to\infty}(1)+N^{-1/6+2\epsilon})\mathcal{M}_{1,A_{4}}.

Now suppose $2\leq d\leq L$ . Let $k=k(d)\geq 1$ denote an integer, with $n_{0}d^{k+1}\leq M$ , to be chosen later. Using Definition 7.8, let

\begin{split}\Sigma_{13,0}^{\bm{a},n_{0}}(d)&\colonequals\sum_{\mathcal{R}\in\mathscr{E}(d,k)}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap(\bm{a}+n_{0}\mathbb{Z}^{m})}a^{\prime}_{\bm{c}}(d)\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1}),\\ \Sigma_{13,1}^{\bm{a},n_{0}}(d)&\colonequals\sum_{\mathcal{R}\in\mathscr{P}(d,k)\setminus\mathscr{E}(d)}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap(\bm{a}+n_{0}\mathbb{Z}^{m})}a^{\prime}_{\bm{c}}(d)\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1}).\end{split}

Then $\Sigma_{13}^{\bm{a},n_{0}}(d)=\Sigma_{13,0}^{\bm{a},n_{0}}(d)+\Sigma_{13,1}^{\bm{a},n_{0}}(d)$ by (7.27) (since $\mathscr{P}(d,k)$ is a partition of $\mathbb{Z}^{m}$ ). By Proposition 7.10, we have $q_{\mathcal{R}}\leq d^{k}$ for all $\mathcal{R}\in\mathscr{P}(d,k)$ , so $Q(\mathscr{P}(d,k))\leq d^{k}$ .

By the triangle inequality,

(7.35)

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,0}^{\bm{a},n_{0}}(d)\rvert\leq\sum_{\mathcal{R}\in\mathscr{E}(d,k)}\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}}\lvert a^{\prime}_{\bm{c}}(d)\rvert\Bigl{\lvert}{\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1})}\Bigr{\rvert}.

By (7.21), $a^{\prime}_{\bm{c}}(d)\ll_{\epsilon}d^{\epsilon}$ . However, by Lemma 5.2 (with $k=1$ and $a(n_{1})=a_{\bm{c},1}(n_{1})$ , and $f(r_{1})=\nu_{\bm{c}/Z}(r_{1}d/N)$ ) and Proposition 5.1, we have (cf. (7.28), (7.29), and (7.30))

\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1})\ll_{b}\int_{1/2}^{AN/d}d^{\times}{N}\int_{t_{1}\in\mathbb{R}}dt_{1}\,\mathcal{M}_{1,b}(\nu)\frac{\lvert\Sigma_{12,(N_{1},1,1)}^{\bm{c},1}(t_{1},0,0)\rvert}{(1+\lvert t_{1}\rvert)^{b}}

for all integers $b\geq 0$ . Plugging this and $a^{\prime}_{\bm{c}}(d)\ll_{\epsilon}d^{\epsilon}$ into (7.35), and then applying Cauchy–Schwarz over $\bm{c}\in\bigcup_{\mathcal{R}\in\mathscr{E}(d,k)}(\mathcal{R}\cap[-Z,Z]^{m})$ , we get (by Lemma 7.12 and Conjecture 6.9)

(7.36)

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,0}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}d^{\epsilon}\mathcal{M}_{1,\lceil A_{3}/2+2\rceil}(\nu)\cdot o_{d;k\to\infty}(Z^{m})^{1/2}\cdot(Z^{m}N/d)^{1/2};

cf. the numerics in (7.31) and (7.32). (Before applying Lemma 7.12, note that $Q(\mathscr{E}(d,k))\leq d^{k}\leq M\leq Z$ , so $\lvert\bigcup_{\mathcal{R}\in\mathscr{E}(d,k)}(\mathcal{R}\cap[-Z,Z]^{m})\rvert\ll Z^{m}\sum_{\mathcal{R}\in\mathscr{E}(d,k)}\mu(\mathcal{R})$ .)

By Definition 7.8, the quantity $a^{\prime}_{\bm{c}}(d)$ is constant over $\mathcal{S}_{1}\cap\mathcal{R}$ for each $\mathcal{R}\in\mathscr{P}(d,k)\setminus\mathscr{E}(d)$ . But $a^{\prime}_{\bm{c}}(d)\ll_{\epsilon}d^{\epsilon}$ by (7.21), so we get

\Sigma_{13,1}^{\bm{a},n_{0}}(d)\ll_{\epsilon}d^{\epsilon}\sum_{\mathcal{R}\in\mathscr{P}(d,k)\setminus\mathscr{E}(d)}\,\Bigl{\lvert}{\sum_{\bm{c}\in\mathcal{S}_{1}\cap\mathcal{R}\cap(\bm{a}+n_{0}\mathbb{Z}^{m})}\sum_{n_{1}\geq 1}\nu_{\bm{c}/Z}(n_{1}d/N)a_{\bm{c},1}(n_{1})}\Bigr{\rvert}.

We may apply Conjecture 6.12 (with $N/d$ in place of $N$ ) and Lemma 6.15 (with $\mathcal{S}=\mathcal{S}_{1}$ and $\theta=m$ ) with $\mathscr{P}=\{\mathcal{R}\cap(\bm{a}+n_{0}\mathbb{Z}^{m}):\mathcal{R}\in\mathscr{P}(d,k),\;1\leq\bm{a}\leq n_{0}\}$ , since $Q(\mathscr{P})\leq n_{0}d^{k}\leq M/d$ and $Z,N\geq 2M$ (so that $2M,Z,N/d\geq 2Q(\mathscr{P})$ and $N/d\leq N\leq Z^{3}$ ). By the triangle inequality applied to the right-hand side of the previous display, we then obtain the bound

(7.37)

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,1}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}d^{\epsilon}\mathfrak{B}_{0}(N/d,\epsilon)\quad\textnormal{(where $\mathfrak{B}_{0}(N,\epsilon)$ denotes \eqref{EXPR:frakB_0-soft-bound-for-Sigma_13(1)-total} as before)}.

Let $K\geq 1$ denote an integer to be chosen soon. Assembling (7.26), (7.32), and our work on $\Sigma_{13}^{\bm{a},n_{0}}(d)$ for $d\leq L$ (see (7.33) for $d=1$ , and (7.36), (7.37) for $2\leq d\leq L$ ), we get (by the triangle inequality) that the sum $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}\rvert$ is at most

(7.38)

\ll_{I,\epsilon}\mathcal{M}_{1,\lceil A_{3}/2+5\rceil}(\nu)(Z^{m}N^{1/2}L^{-\eta_{4}}+o_{L;K\to\infty}(Z^{m}N^{1/2}))+\sum_{d\leq L}d^{\epsilon}\mathfrak{B}_{0}(N/d,\epsilon),

provided $k(d)\geq K$ and $n_{0}d^{k(d)+1}\leq M$ hold for all integers $d$ with $2\leq d\leq L$ . But upon replacing $N$ in (7.33) with $N/d$ , and summing over $d\leq L$ , we find that

(7.39)

\sum_{d\leq L}d^{\epsilon}\mathfrak{B}_{0}(N/d,\epsilon)\ll L^{2/3}\mathfrak{B}_{0}(N,\epsilon).

It remains to carefully specify parameters. Choose $K=K(L)\geq 1$ so that $o_{L;K\to\infty}(Z^{m}N^{1/2})\leq L^{-\eta_{4}}Z^{m}N^{1/2}$ . Then let $k(d)=K(L)$ for all integers $d$ with $2\leq d\leq L$ . Let $L=L(M)\geq 1$ (in terms of $M$ ) be the largest integer for which $L^{K(L)+1}\leq M^{1/10}$ ; such an integer exists, because $1^{K(1)+1}\leq M^{1/10}$ and $L^{K(L)+1}\geq L^{2}$ . Crucially, we have $\lim_{M\to\infty}{L(M)}=\infty$ , since the expression $L^{K(L)+1}$ is bounded on any finite set of integers $L$ .

Since $n_{0}\leq M^{9/10}$ , we have $n_{0}L^{K(L)+1}\leq M$ . So the bound (7.38) on $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}\rvert$ is valid, and therefore (by (7.39)) we get (letting $A_{7}=\max(A_{4},\lceil A_{3}/2+5\rceil)$ )

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}\rvert\ll_{I,\epsilon}\mathcal{M}_{1,A_{7}}(\nu)Z^{m}N^{1/2}L(M)^{-\eta_{4}}+L(M)^{2/3}\mathfrak{B}_{0}(N,\epsilon).

But $L(M)^{2}\leq L(M)^{K(L(M))+1}\leq M^{1/10}$ , and $N\geq M$ , so by (7.34), we have

L(M)^{2/3}\mathfrak{B}_{0}(N,\epsilon)\ll Z^{m}N^{1/2}(M^{1/30-1/6+2\epsilon}+o_{M;Z\to\infty}(1))\mathcal{M}_{1,A_{7}},

provided $\epsilon$ is sufficiently small. Both $L(M)^{-\eta_{4}}$ and $M^{1/30-1/6+2\epsilon}$ are $o_{M\to\infty}(1)$ . ∎

Proposition 7.16 (RA1 $\delta$ ’E’).

Assume Conjectures 1.2, 1.10, and 1.11. Suppose $Z,N\geq 2M$ and $N\leq Z^{3}$ . Suppose $1\leq M\leq Z^{\eta_{2}}$ (with $\eta_{2}$ as in Proposition 6.14), and let $n_{0}\leq M^{1/2}$ be a positive integer. Then $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}(\nu,Z,N)\rvert\ll_{F,I,\epsilon}Z^{m+\epsilon}N^{1/2}\cdot(M^{1/6\deg{H}}N^{-1/6}+M^{-\eta_{5}/4\deg{H}})\cdot\mathcal{M}_{1,A_{8}}$ . Here $\eta_{5}$ , $A_{8}$ are positive reals depending only on $F$ .

Proof.

We adjust the proof of Proposition 7.15. Let $L\in[1,M^{1/8}]$ be a real to be chosen later. For each integer $d$ with $2\leq d\leq L$ , let $k(d)$ be the largest integer $k$ for which $d^{k+1}\leq M^{1/2}$ ; then $k(d)\geq 3$ (since $d\leq L\leq M^{1/8}$ ), and $d^{k(d)+2}>M^{1/2}$ (by the maximality of $k(d)$ ). Let

\mathfrak{B}_{1}(N,\epsilon)\colonequals Z^{m-\eta_{2}}N^{1/2}\mathcal{M}_{1,A_{5}}+Z^{m}N^{1/3+\epsilon}M^{\epsilon}\mathcal{M}_{1,0}.

Using Proposition 6.14 in place of Conjecture 6.12, we find that $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13}^{\bm{a},n_{0}}(1)\rvert\ll_{I,\epsilon}\mathfrak{B}_{1}(N,\epsilon)$ and (if $2\leq d\leq L$ , then) $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,1}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}d^{\epsilon}\mathfrak{B}_{1}(N/d,\epsilon)$ ; cf. (7.33) and (7.37). Summing over $1\leq d\leq L$ , and writing $\Sigma_{13,1}^{\bm{a},n_{0}}(1)\colonequals\Sigma_{13}^{\bm{a},n_{0}}(1)$ for convenience, we obtain

(7.40)

\sum_{1\leq d\leq L}\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,1}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}L^{2/3}\mathfrak{B}_{1}(N,\epsilon)\leq L^{2/3}Z^{m+4\epsilon}N^{1/2}(Z^{-\eta_{2}}+N^{-1/6})\mathcal{M}_{1,A_{5}}.

On the other hand, if we plug GRH (Proposition 3.2(8)) into (7.35) (after partial summation over $n_{1}$ , say, and recalling the definition of $\mathcal{M}_{1,1}$ from (6.24)) and then use Lemma 7.13 (applicable since $k(d)\geq 3$ and we assume Conjecture 1.11), then (if $2\leq d\leq L$ ) we get

(7.41)

\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,0}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}d^{\epsilon}\mathcal{M}_{1,1}(\nu)Z^{\epsilon}(N/d)^{1/2+\epsilon}\cdot Z^{m}d^{k(d)\epsilon}d^{-(k(d)-3)/\deg{H}};

cf. the use of Lemma 7.12 towards (7.36). On the right-hand side, $d^{\epsilon}Z^{\epsilon}(N/d)^{\epsilon}d^{k(d)\epsilon}\leq Z^{5\epsilon}$ (since $N\leq Z^{3}$ and $d^{k(d)}\leq M$ ), and $d^{(k(d)-3)/\deg{H}}>M^{1/2\deg{H}}d^{-5/\deg{H}}\geq M^{1/2\deg{H}}d^{-5/12}$ (since $d^{k(d)+2}>M^{1/2}$ and $\deg{H}\geq 12$ ). Thus, summing (7.41) over $2\leq d\leq L$ gives

(7.42)

\sum_{2\leq d\leq L}\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{13,0}^{\bm{a},n_{0}}(d)\rvert\ll_{I,\epsilon}\mathcal{M}_{1,1}(\nu)Z^{m+5\epsilon}N^{1/2}L^{11/12}\cdot M^{-1/2\deg{H}}.

As for $\Sigma_{12}^{\bm{a},n_{0}}$ , the bounds (7.31) and (7.32) must be adjusted slightly, since we do not assume Conjecture 6.9. However, by Proposition 3.2(8) and partial summation, the bound (6.23) in Conjecture 6.9 still holds up to a factor of $Z^{\epsilon}$ (for any $A_{3}>0$ ). Therefore, (7.31) and (7.32) still hold if we replace $Z^{m}N^{1/2}$ with $Z^{m+\epsilon}N^{1/2}$ ; so $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{12}^{\bm{a},n_{0}}\rvert\ll_{I}\mathcal{M}_{1,\lceil A_{3}/2+5\rceil}(\nu)Z^{m}N^{1/2}/L^{\eta_{4}}$ .

Let $A_{8}=\max(A_{5},\lceil A_{3}/2+5\rceil)$ . Assembling (7.26), (7.40), (7.42), and our work on $\Sigma_{12}^{\bm{a},n_{0}}$ , we get (by the triangle inequality) that the sum $\sum_{1\leq\bm{a}\leq n_{0}}\lvert\Sigma_{11}^{\bm{a},n_{0}}\rvert$ is

\ll_{I,\epsilon}\mathcal{M}_{1,A_{8}}(\nu)Z^{m+5\epsilon}N^{1/2}(L^{2/3}(Z^{-\eta_{2}}+N^{-1/6})+L^{11/12}M^{-1/2\deg{H}}+L^{-\eta_{4}}).

Let $L=M^{1/4\deg{H}}$ and $\eta_{5}=\min(1,\eta_{4})$ to finish. ∎

8. New bounds on the integral factor

Recall $J_{\bm{c},X}(n)$ from (2.9). As we explained in §2, we need to go beyond the integral estimates from standard sources like [duke1993bounds, heath1996new, heath1998circle, hooley2014octonary] (and the related estimates of [hooley1986HasseWeil, hooley_greaves_harman_huxley_1997]), such as (2.19). We will prove uniform bounds free of epsilons and logs, while also bringing discriminants into the picture via the following consequence of (2.3):

(8.1)

\Delta(\nabla{F}(\bm{x}))\ll_{F}\lvert F(\bm{x})\rvert\cdot\lVert\bm{x}\rVert^{2\deg(\Delta)-\deg(F)}\quad\textnormal{for $\bm{x}\in\mathbb{R}^{m}$}.

To give clean, general bounds, we assume (2.6). We prove the following on $J_{\bm{c},X}(n)$ :

Proposition 8.1.

Assume (2.6). Let $X,Z,n\in\mathbb{R}_{>0}$ . Let $\bm{c}\in\mathbb{R}^{m}$ with $\lVert\bm{c}\rVert\leq Z$ . Then

\partial_{\log{n}}^{j}\partial_{\bm{c}}^{\bm{\alpha}}J_{\bm{c},X}(n)\ll_{F,w,j,\bm{\alpha},b}\frac{(X/n)^{\lvert\bm{\alpha}\rvert}(1+X\lVert\bm{c}\rVert/n)^{1-m/2}}{(1+\lVert\bm{c}\rVert/X^{1/2})^{b}(1+X\lVert\Delta(\bm{c}/Z)\bm{c}\rVert/n)^{b}}

for all integers $j,b\geq 0$ and multi-indices $\bm{\alpha}\geq 0$ .

The fact that increasing $j$ is harmless can be interpreted as an instance of “homogeneous dimensional analysis” (and ultimately arises from the homogeneity of $F$ , via a beautiful recursive structure due to [heath1996new]; see (8.5) below). The factor $\Delta(\bm{c}/Z)\ll 1$ measures the “degeneracy” of the real hyperplane section $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ , and it arises in our proof for roughly the same reason that dual hypersurfaces arise in [huang2020density]*(5.4), (5.11)–(5.14).

Morally, in (2.10), Proposition 8.1 lets us “imagine that there are sharp cutoffs” $\lVert\bm{c}\rVert\ll X^{1/2}$ and $n\gg X\lVert\Delta(\bm{c}/X^{1/2})\bm{c}\rVert$ . Since by (4.4) we typically have $\lvert\Delta(\bm{c}/X^{1/2})\rvert\asymp 1$ , one might thus expect (in view of Proposition 2.4, and our $\epsilon$ -diagnosis at the end of §2) that $n\asymp X^{3/2}$ should be the “dominant range” on average, and there we have $J_{\bm{c},X}(n)\ll 1$ .

To prove Proposition 8.1, we must first dig into some technical aspects of [duke1993bounds, heath1996new]’s $h$ -function. Let $\omega_{\textnormal{HB},0}(x)\colonequals\bm{1}_{\lvert x\rvert\leq 1}\cdot\exp(-(1-x^{2})^{-1})\in C^{\infty}_{c}(\mathbb{R})$ and $c_{\textnormal{HB},0}\colonequals\int_{x\in\mathbb{R}}dx\,\omega_{\textnormal{HB},0}(x)$ ; note that $\omega_{\textnormal{HB},0}$ is supported on $[-1,1]$ . Following [heath1996new]*p. 165, let

\omega_{\textnormal{HB}}(x)\colonequals 4c_{\textnormal{HB},0}^{-1}\cdot\omega_{\textnormal{HB},0}(4x-3)\in C^{\infty}_{c}(\mathbb{R}),

so that $\omega_{\textnormal{HB}}$ is supported on $[1/2,1]$ . For $x>0$ and $y\in\mathbb{R}$ , let

h(x,y)\colonequals\sum_{j\geq 1}(xj)^{-1}[\omega_{\textnormal{HB}}(xj)-\omega_{\textnormal{HB}}(\lvert y\rvert/(xj))].

By (2.7) and (2.9), and a change of variables from $\bm{x}$ to $\tilde{\bm{x}}=\bm{x}/X$ , we have (since $Y^{2}=X^{3}$ )

(8.2)

J_{\bm{c},X}(n)=\int_{\tilde{\bm{x}}\in\mathbb{R}^{m}}d\tilde{\bm{x}}\,w(\tilde{\bm{x}})h(n/Y,F(\tilde{\bm{x}}))e(-X\bm{c}\cdot\tilde{\bm{x}}/n).

The following shows that we may take $A_{0}=\sup_{\bm{x}\in\operatorname{Supp}{w}}\max(1,2\lvert F(\bm{x})\rvert)$ in Proposition 2.4.

Lemma 8.2 (See [heath1996new]*Lemma 4).

If $\xi\in\mathbb{R}$ and $r\geq\max(1,2\lvert\xi\rvert)$ , then $h(r,\xi)=0$ .

Following [heath1996new], we now build a Fourier transform. Fix $\upsilon_{1}\in C^{\infty}_{c}(\mathbb{R})$ such that $\upsilon_{1}(F(\bm{x}))\geq 1$ for all $\bm{x}\in\operatorname{Supp}{w}$ . Let $w_{0}(\bm{x})\colonequals w(\bm{x})/\upsilon_{1}(F(\bm{x}))\in C^{\infty}_{c}(\mathbb{R})$ . For any $r>0$ , let

p_{r}(u)\colonequals\int_{\xi\in\mathbb{R}}d\xi\,\upsilon_{1}(\xi)h(r,\xi)e(-u\xi)

be the Fourier transform of $\upsilon_{1}(\xi)h(r,\xi)\in C^{\infty}_{c}(\mathbb{R})$ . Writing $w(\bm{x})=w_{0}(\bm{x})\upsilon_{1}(F(\bm{x}))$ and $\upsilon_{1}(\xi)h(r,\xi)=\int_{u\in\mathbb{R}}du\,p_{r}(u)e(u\xi)$ (for $r=n/Y$ and $\xi=F(\bm{x})$ ) in (8.2), we get

(8.3)

J_{\bm{c},X}(n)=\int_{u\in\mathbb{R}}du\,p_{r}(u)\int_{\tilde{\bm{x}}\in\mathbb{R}^{m}}d\tilde{\bm{x}}\,w_{0}(\tilde{\bm{x}})e(uF(\tilde{\bm{x}})-\bm{v}\cdot\tilde{\bm{x}}),

where $(r,\bm{v})=(n/Y,X\bm{c}/n)$ ; cf. [heath1996new]*Lemma 17.

Let $\mathfrak{q}_{r}(t)\colonequals p_{r}(t/r)$ , so that $p_{r}(u)=\mathfrak{q}_{r}(ru)$ . It turns out (see Lemma 8.3) that $\mathfrak{q}_{r}$ behaves somewhat like a “fixed” Schwartz function independent of $r$ . Thus $u$ may be compared with $\beta Y^{2}$ in the classical Dirichlet arc theory, where $\lvert\beta\rvert\leq 1/(nY)=1/(rY^{2})$ .

Lemma 8.3.

Let $r>0$ and $t\in\mathbb{R}$ . Then $\partial_{\log{r}}^{j}\mathfrak{q}_{r}^{(l)}(t)\ll_{l,j,k}(1+\lvert t\rvert)^{-k}$ for all $j,k,l\in\mathbb{Z}_{\geq 0}$ .

Proof.

We may assume $k=0$ if $\lvert t\rvert\leq 1$ ; this lets us treat all $t$ simultaneously.

Since $\mathfrak{q}_{r}(t)=p_{r}(t/r)=\int_{\xi\in\mathbb{R}}d\xi\,\upsilon_{1}(\xi)h(r,\xi)e(-t\xi/r)$ , we have

\mathfrak{q}_{r}^{(l)}(t)=\int_{\xi\in\mathbb{R}}d\xi\,\upsilon_{1}(\xi)h(r,\xi)\cdot(-2\pi i\xi/r)^{l}e(-t\xi/r).

Since $\partial_{\log{r}}((\xi/r)^{l})=-l(\xi/r)^{l}$ and $\partial_{\log{r}}(e(-t\xi/r))=(2\pi it\xi/r)e(-t\xi/r)$ , we find by induction (and the Leibniz rule) that for some constants $c^{l,j}_{a,b}\in\mathbb{C}$ , we have

\partial_{\log{r}}^{j}\mathfrak{q}_{r}^{(l)}(t)=\sum_{a,b\geq 0:\,a+b\leq j}c^{l,j}_{a,b}\int_{\xi\in\mathbb{R}}d\xi\,\upsilon_{1}(\xi)\cdot(\partial_{\log{r}}^{a}h(r,\xi))\cdot(\xi/r)^{l}(t\xi/r)^{b}e(-t\xi/r).

Integrating by parts $k$ times in $\xi$ (repeatedly integrating $e(-t\xi/r)$ and differentiating the complementary factor), and then taking absolute values, we get

\partial_{\log{r}}^{j}\mathfrak{q}_{r}^{(l)}(t)\ll_{l,j,k,\upsilon_{1}}\sum_{\begin{subarray}{c}a,b,\alpha,\beta\geq 0:\\ a+b\leq j,\;\alpha+\beta\leq k,\;\beta\leq l+b\end{subarray}}\int_{\xi\in\operatorname{Supp}{\upsilon_{1}}}d\xi\,(\partial_{\xi}^{\alpha}\partial_{\log{r}}^{a}h(r,\xi))\cdot\frac{\lvert\xi/r\rvert^{l}\lvert t\xi/r\rvert^{b}}{\lvert\xi\rvert^{\beta}}\frac{1}{\lvert t/r\rvert^{k}}.

By Lemma 8.2, we may assume $r\ll_{\upsilon_{1}}1$ . Then $\partial_{r}^{c}\partial_{\xi}^{\alpha}h(r,\xi)\ll_{c,\alpha,A}r^{-1-c-\alpha}\min(1,(r/\lvert\xi\rvert)^{A})$ for $c,\alpha,A\geq 0$ , by [heath1996new]*Lemma 5. Thus $\partial_{\log{r}}^{a}\partial_{\xi}^{\alpha}h(r,\xi)\ll_{a}\sum_{0\leq c\leq a}r^{c}\cdot\lvert\partial_{r}^{c}\partial_{\xi}^{\alpha}h(r,\xi)\rvert\ll_{a,\alpha,A}r^{-1-\alpha}\min(1,(r/\lvert\xi\rvert)^{A})$ , which when inserted in the previous display gives

\partial_{\log{r}}^{j}\mathfrak{q}_{r}^{(l)}(t)\ll_{l,j,k,A}\sum_{\begin{subarray}{c}a,b,\alpha,\beta\geq 0:\\ a+b\leq j,\;\alpha+\beta\leq k,\;\beta\leq l+b\end{subarray}}\int_{\xi\in\mathbb{R}}d\xi\,\frac{r^{-1-\alpha-l-b+k}\lvert\xi\rvert^{l+b-\beta}\lvert t\rvert^{b-k}}{(1+\lvert\xi/r\rvert)^{A}}.

Let $A=l+j+2$ ; then each integral here over $\xi$ is $\ll_{l,j,k}r^{-1-\alpha-l-b+k}r^{1+l+b-\beta}\lvert t\rvert^{b-k}$ (by Lemma 5.3), and thus $\ll_{l,j,k}r^{k-\alpha-\beta}\lvert t\rvert^{b-k}\ll\lvert t\rvert^{b-k}$ (since $k-\alpha-\beta\geq 0$ and $r\ll 1$ ). If $\lvert t\rvert\leq 1$ , we are done (since $k=0$ ). If $\lvert t\rvert>1$ , we may replace $k$ with $k+j$ to finish (since $b\leq j$ ). ∎

We now put (8.3) in a broader framework. For any Schwartz functions $q\colon\mathbb{R}\to\mathbb{C}$ and $\phi\in C^{\infty}_{c}(\mathbb{R}^{m})\otimes\mathbb{C}$ with $\operatorname{Supp}{\phi}\subseteq\operatorname{Supp}{w}$ , let

(8.4)

\mathscr{J}_{r,\bm{v}}(q,\phi)\colonequals\int_{u\in\mathbb{R}}du\,q(ru)\int_{\bm{x}\in\mathbb{R}^{m}}d\bm{x}\,\phi(\bm{x})e(uF(\bm{x})-\bm{v}\cdot\bm{x}).

Then by (8.3), we have $J_{\bm{c},X}(n)=\mathscr{J}_{r,\bm{v}}(\mathfrak{q}_{r},w_{0})$ , where $(r,\bm{v})=(n/Y,X\bm{c}/n)$ .

Proposition 8.4.

Assume (2.6). Let $q$ , $\phi$ be as above (with $q$ , $\phi$ Schwartz and $\operatorname{Supp}\phi\subseteq\operatorname{Supp}{w}$ ). Then for all $(k,\bm{v})\in\mathbb{Z}_{\geq 0}\times\mathbb{R}^{m}$ and positive reals $r\leq A_{0}$ and $M\geq\lVert\bm{v}\rVert$ , we have

\mathscr{J}_{r,\bm{v}}(q,\phi)\ll_{F,w,q,\phi,k}(1+\lVert\bm{v}\rVert)^{1-m/2}\cdot(1+\lVert r\bm{v}\rVert)^{-k}(1+\lVert\Delta(\bm{v}/M)\bm{v}\rVert)^{-k}.

Before proving Proposition 8.4, we first explain why it implies the desired Proposition 8.1. In order to handle $\partial_{\log{n}}^{\geq 1}$ , we need a recursion, (8.5), originally observed to first order by [heath1996new]. Without such a recursion, we might suffer for small moduli $n$ , as in [hooley1986HasseWeil]*§9’s analysis of “junior arcs” (repaired in [hooley_greaves_harman_huxley_1997] for some purposes, by a clever averaging argument).

Lemma 8.5.

Let $(r,\bm{v})=(n/Y,X\bm{c}/n)$ . Let $q_{1}(t)\colonequals t\cdot q(t)$ and let $\phi_{1}(\bm{x})\colonequals\operatorname{div}(\phi_{1}(\bm{x})\bm{x})=\bm{x}\cdot\nabla{\phi_{1}}(\bm{x})=\sum_{1\leq j\leq m}x_{j}\cdot\partial_{x_{j}}{\phi_{1}}$ . Let $\phi_{2,j}(\bm{x})\colonequals x_{j}\cdot\phi(\bm{x})$ . Then

(8.5)		$\displaystyle\partial_{\log{n}}\mathscr{J}_{r,\bm{v}}(q,\phi)$	$\displaystyle=(m-3)\mathscr{J}_{r,\bm{v}}(q,\phi)-2\mathscr{J}_{r,\bm{v}}(q_{1},\phi)+\mathscr{J}_{r,\bm{v}}(q,\phi_{1}),$
(8.6)		$\displaystyle\partial_{c_{j}}\mathscr{J}_{r,\bm{v}}(q,\phi)$	$\displaystyle=(-2\pi iX/n)\mathscr{J}_{r,\bm{v}}(q,\phi_{2,j}).$

Proof.

The formula (8.6) immediately follows upon differentiating (8.4) by $c_{j}$ . It is possible to prove (8.5) by a clever integration by parts (cf. [heath1996new]*p. 182, proof of Lemma 14). We give a slightly shorter argument. Write $u=t/r^{3}$ and $\bm{x}=r\bm{y}$ in (8.4) to get

(8.7)

\mathscr{J}_{r,\bm{v}}(q,\phi)=r^{m-3}\int_{\mathbb{R}}dt\,q(t/r^{2})\int_{\mathbb{R}^{m}}d\bm{y}\,\phi(r\bm{y})e(tF(\bm{y})-r\bm{v}\cdot\bm{y}).

Differentiating both sides of (8.7) by $\log{n}$ , using $\partial_{\log{n}}{r}=r$ and the fact that $r\bm{v}=X\bm{c}/Y$ is independent of $\log{n}$ , we get (8.5), by (8.7) applied to each of $(q,\phi)$ , $(q_{1},\phi)$ , $(q,\phi_{1})$ . ∎

Proof of Proposition 8.1, assuming Proposition 8.4.

Recall that by (8.3), we have $J_{\bm{c},X}(n)=\mathscr{J}_{r,\bm{v}}(\mathfrak{q}_{r},w_{0})$ , where $(r,\bm{v})=(n/Y,X\bm{c}/n)$ . By Lemmas 8.3 and 8.5 (first using the chain rule, (8.5), and Lemma 8.3 when differentiating by $\log{n}$ , and then using (8.6) when differentiating $\bm{c}$ ), we may thus write $\partial_{\bm{c}}^{\bm{\alpha}}\partial_{\log{n}}^{j}J_{\bm{c},X}(n)$ as a finite linear combination of integrals $\mathscr{J}_{r,\bm{v}}(q,\phi)$ , with coefficients $O_{m,j,\bm{\alpha}}((X/n)^{\lvert\bm{\alpha}\rvert})$ , running over a set of at most $4^{j}$ pairs $(q,\phi)$ . (For example, for $j=1$ and $\bm{\alpha}=\bm{0}$ , we would use the chain rule and (8.5) to write $\partial_{\log{n}}\mathscr{J}_{r,\bm{v}}(\mathfrak{q}_{r},w_{0})$ as

\mathscr{J}_{r,\bm{v}}(\partial_{\log{r}}{\mathfrak{q}_{r}},w_{0})+(m-3)\mathscr{J}_{r,\bm{v}}(\mathfrak{q}_{r},w_{0})-2\mathscr{J}_{r,\bm{v}}(\mathfrak{q}_{r,1},w_{0})+\mathscr{J}_{r,\bm{v}}(q,w_{0,1}),

where $\mathfrak{q}_{r,1}(t)=t\cdot\mathfrak{q}_{r}(t)$ and $w_{0,1}(\bm{x})\colonequals\operatorname{div}(w_{0}(\bm{x})\bm{x})$ .) Proposition 8.1 then immediately follows from Proposition 8.4 (applied to each individual $\mathscr{J}_{r,\bm{v}}(q,\phi)$ ). ∎

To prove Proposition 8.4, we need the following lemma; the basic principle is familiar (see e.g. [hormander1990analysis]*Theorem 7.7.1) but the treatment of [heath1996new] is ideal for us.

Lemma 8.6 (Non-stationary phase).

Let $\lambda,A\in\mathbb{R}_{>0}$ and $d,k\in\mathbb{Z}_{\geq 1}$ . Suppose $a\in C^{\infty}_{c}(\mathbb{R}^{d})\otimes\mathbb{C}$ is supported on $[-A,A]^{d}$ , with $\sum_{\lvert\bm{\alpha}\rvert\leq k}\lvert\partial_{\bm{x}}^{\bm{\alpha}}{a}(\bm{x})\rvert\leq A$ for all $\bm{x}\in\mathbb{R}^{d}$ . Suppose $f\colon\mathbb{R}^{d}\to\mathbb{R}$ is smooth, with $\lVert\nabla{f}(\bm{x})\rVert\geq\lambda/A$ and $\sum_{2\leq\lvert\bm{\alpha}\rvert\leq k+1}\lvert\partial_{\bm{x}}^{\bm{\alpha}}{f}(\bm{x})\rvert\leq A\lambda$ for all $\bm{x}\in\operatorname{Supp}{a}$ . Then

\int_{\bm{x}\in\mathbb{R}^{d}}d\bm{x}\,a(\bm{x})e(f(\bm{x}))\ll_{d,A,k}\lambda^{-k}.

Proof.

See [heath1996new]*Lemma 10 and its proof (repeated integration by parts); see [heath1996new]*§2 for the definition of $\mathscr{C}(S)$ (which allows for the required uniformity over weight functions). Note that we do not require any explicit upper bound on $\sup_{\bm{x}\in\operatorname{Supp}{a}}{\lvert f(\bm{x})\rvert}$ , or any control on the shape of the compact set $\operatorname{Supp}{a}\subseteq[-A,A]^{d}$ . ∎

Proof of Proposition 8.4.

Certainly $\bm{0}\notin\operatorname{Supp}{w}$ by (2.6). Since $V$ is smooth, we thus have $\lVert\nabla{F}(\bm{x})\rVert>0$ for all $\bm{x}\in\operatorname{Supp}{w}$ . Since $\operatorname{Supp}{w}$ is compact, there thus exists $A_{9}=A_{9}(F,w)\geq 2$ such that for all $\bm{x}\in\operatorname{Supp}{w}$ , we have $\lVert\bm{x}\rVert,\lVert\nabla{F}(\bm{x})\rVert\in[A_{9}^{-1},A_{9}]$ . Let $\mathscr{W}\colonequals\{a\bm{x}:(a,\bm{x})\in[1/3A_{9},3A_{9}]\times\operatorname{Supp}{w}\}$ be the union of all dilates of $\operatorname{Supp}{w}$ by a scale factor $a\in[1/3A_{9},3A_{9}]$ . The set $\mathscr{W}$ is compact, being the continuous image of a product of compact sets. Also, since the right-hand side of (2.6) is invariant under scaling, we have

(8.8)

\mathscr{W}\subseteq\{\bm{w}\in\mathbb{R}^{m}:\det(\operatorname{Hess}{F}(\bm{w}))\neq 0\}.

Let $A_{10}=A_{10}(F,w)\geq 2$ be a constant such that for all $\bm{w}\in\mathscr{W}$ , we have

(8.9)

A_{10}^{-1}\leq\lVert\bm{w}\rVert\leq A_{10},\quad A_{10}^{-1}\leq\lVert\nabla{F}(\bm{w})\rVert\leq A_{10}.

Our plan is to first consider the $\bm{x}$ -integral in $\mathscr{J}_{r,\bm{v}}(q,\phi)$ (see (8.4)) for a single value of $u$ at a time, and then integrate over $u$ . Given $u$ , $\bm{v}$ , let $\psi_{0}(\bm{x})\colonequals uF(\bm{x})-\bm{v}\cdot\bm{x}$ and $\mathcal{J}_{u,\bm{v}}(\phi)\colonequals\int_{\bm{x}\in\mathbb{R}^{m}}d\bm{x}\,\phi(\bm{x})e(\psi_{0}(\bm{x}))$ . Taking absolute values in $\mathcal{J}$ gives the trivial bound

(8.10)

\mathcal{J}_{u,\bm{v}}(\phi)\ll_{\phi}1.

On the other hand, $\nabla{\psi_{0}}(\bm{x})=u\nabla{F}(\bm{x})-\bm{v}$ . In particular, if $\lvert u\rvert\geq 2A_{9}\lVert\bm{v}\rVert$ or $\lvert u\rvert\leq\lVert\bm{v}\rVert/2A_{9}$ , then we have $\lVert\nabla{\psi_{0}}(\bm{x})\rVert\geq\max(\lvert u/2A_{9}\rvert,\lVert\bm{v}\rVert/2)$ and $\partial_{\bm{x}}^{\bm{\alpha}}{\psi_{0}}(\bm{x})\ll_{F,w,\bm{\alpha}}\lvert u\rvert+\lVert\bm{v}\rVert$ for all $\bm{x}\in\operatorname{Supp}{w}$ , and thus $\mathcal{J}_{u,\bm{v}}(\phi)\ll_{\phi,k}(\lvert u\rvert+\lVert\bm{v}\rVert)^{-k}$ by Lemma 8.6 (provided $\lvert u\rvert+\lVert\bm{v}\rVert>0$ ). This, together with (8.10) and the bound $q(ru)\ll_{q}1$ , gives

(8.11)		$\displaystyle\int_{\lvert u\rvert\geq 2A_{9}\lVert\bm{v}\rVert}du\,\lvert q(ru)\cdot\mathcal{J}_{u,\bm{v}}(\phi)\rvert$	$\displaystyle\ll_{q,\phi,b}\int_{\lvert u\rvert\geq 2A_{9}\lVert\bm{v}\rVert}\frac{du}{(1+\lvert u\rvert)^{b+1}}\ll_{b}(1+\lVert\bm{v}\rVert)^{-b},$
(8.12)		$\displaystyle\int_{\lvert u\rvert\leq\lVert\bm{v}\rVert/2A_{9}}du\,\lvert q(ru)\cdot\mathcal{J}_{u,\bm{v}}(\phi)\rvert$	$\displaystyle\ll_{q,\phi,b}\int_{\lvert u\rvert\leq\lVert\bm{v}\rVert/2A_{9}}\frac{du}{(1+\lVert\bm{v}\rVert)^{b+1}}\ll_{b}(1+\lVert\bm{v}\rVert)^{-b},$

for all integers $b\geq 1$ . Fix $w_{1}\in C^{\infty}_{c}(\mathbb{R})$ with

w_{1}|_{[-2A_{9},2A_{9}]\setminus[-1/2A_{9},1/2A_{9}]}=1,\quad\operatorname{Supp}{w_{1}}\subseteq[-3A_{9},3A_{9}]\setminus[-1/3A_{9},1/3A_{9}].

Then by (8.11), (8.12), and the triangle inequality, we have (for integers $b\geq 1$ )

(8.13)

\mathscr{J}_{r,\bm{v}}(q,\phi)-\int_{u\in\mathbb{R}}du\,q(ru)w_{1}(u/\lVert\bm{v}\rVert)\mathcal{J}_{u,\bm{v}}(\phi)\ll_{q,\phi,b}(1+\lVert\bm{v}\rVert)^{-b}.

It remains to handle $\lvert u\rvert/\lVert\bm{v}\rVert\in[1/3A_{9},3A_{9}]$ . Suppose first that $\lVert\bm{v}\rVert\leq 1$ . Plugging (8.10) and the bound $q(ru)\ll_{q}1$ into (8.13), we get $\mathscr{J}_{r,\bm{v}}(q,\phi)\ll_{q,\phi}(1+\lVert\bm{v}\rVert)^{-1}+\int_{\lvert u\rvert\leq 3A_{9}}du\,1\ll 1$ . This bound on $\mathscr{J}_{r,\bm{v}}(q,\phi)$ fits in Proposition 8.4, since $\lVert r\bm{v}\rVert\leq A_{0}$ and $\lVert\Delta(\bm{v}/M)\bm{v}\rVert\ll_{F}1$ .

For the rest of the proof, suppose $\lVert\bm{v}\rVert\geq 1$ , let $\tilde{u}\colonequals u/\lVert\bm{v}\rVert$ and $\tilde{\bm{v}}\colonequals\bm{v}/\lVert\bm{v}\rVert$ , let $\bm{z}\colonequals\lvert\tilde{u}\rvert^{1/2}\bm{x}$ , let $\operatorname{sgn}(u)\colonequals u/\lvert u\rvert\in\{-1,1\}$ , and let

\psi_{1}(\bm{z})\colonequals\lvert\tilde{u}\rvert^{1/2}\psi_{0}(\bm{x})/\lVert\bm{v}\rVert=\lvert\tilde{u}\rvert^{1/2}\psi_{0}(\bm{z}/\lvert\tilde{u}\rvert^{1/2})/\lVert\bm{v}\rVert=\operatorname{sgn}(u)F(\bm{z})-\tilde{\bm{v}}\cdot\bm{z}.

(The normalization by $\lvert\tilde{u}\rvert^{1/2}$ makes $\psi_{1}$ nearly independent of $u$ ; this is crucial later.) Note that $\nabla{\psi_{1}}(\bm{z})=\operatorname{sgn}(u)\nabla{F}(\bm{z})-\tilde{\bm{v}}=\tilde{u}\nabla{F}(\bm{x})-\tilde{\bm{v}}$ . Now consider an individual $\tilde{u}\in\mathbb{R}$ with

(8.14)

1/3A_{9}\leq\lvert\tilde{u}\rvert\leq 3A_{9}.

For all $\bm{x}\in\operatorname{Supp}{w}$ , we have $\bm{z}\in\mathscr{W}$ , so $\lVert\bm{z}\rVert\in[1/A_{10},A_{10}]$ and $\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq\lVert\nabla{F}(\bm{z})\rVert+1\leq 2A_{10}$ . Furthermore, the condition (8.8) implies

(8.15)

\operatorname{vol}{\{\bm{z}\in\mathscr{W}:\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda\}}\ll_{F,w}\lambda^{m},

uniformly over $\lambda\in\mathbb{R}_{>0}$ , by calculus (see [heath1996new]*Lemma 21 and its proof). We will split $\mathcal{J}_{u,\bm{v}}(\phi)$ into pieces according to the size of $\lVert\nabla{\psi_{1}}(\bm{z})\rVert$ ; cf. [huang2020density]* $\mathscr{K}_{i}$ on p. 2061. To avoid the need for explicit stationary phase expansions (which can be messy after the leading term), we will also use a subdivision process inspired by [heath1996new]*§8.

Recall the weights $\nu_{0}$ , $\nu_{1}$ from §1.2. Let $\overline{\nu}_{0}\colonequals 1-\nu_{0}$ . For reals $\lambda>0$ , let

(8.16)		$\displaystyle\mathcal{J}_{0,u,\bm{v}}$	$\displaystyle\colonequals\int_{\mathbb{R}^{m}}\frac{d\bm{z}}{\lvert\tilde{u}\rvert^{m/2}}\,\nu_{0}{\left(\frac{\nabla{\psi_{1}}(\bm{z})}{\lVert\bm{v}\rVert^{-1/2}}\right)}\phi(\bm{x})e{\left(\frac{\lVert\bm{v}\rVert\psi_{1}(\bm{z})}{\lvert\tilde{u}\rvert^{1/2}}\right)},$
(8.17)		$\displaystyle\mathcal{J}_{1,u,\bm{v},\lambda}$	$\displaystyle\colonequals\int_{\mathbb{R}^{m}}\frac{d\bm{z}}{\lvert\tilde{u}\rvert^{m/2}}\,\overline{\nu}_{0}{\left(\frac{\nabla{\psi_{1}}(\bm{z})}{\lVert\bm{v}\rVert^{-1/2}}\right)}\nu_{1}{\left(\frac{\nabla{\psi_{1}}(\bm{z})}{\lambda}\right)}\phi(\bm{x})e{\left(\frac{\lVert\bm{v}\rVert\psi_{1}(\bm{z})}{\lvert\tilde{u}\rvert^{1/2}}\right)},$

where $\phi(\bm{x})=\phi(\bm{z}/\lvert\tilde{u}\rvert^{1/2})$ . By the definitions of $\nu_{0}$ , $\nu_{1}$ , we have $\mathcal{J}_{1,u,\bm{v},\lambda}=0$ unless there exists $\bm{z}\in\mathscr{W}$ (corresponding to $\bm{x}\in\operatorname{Supp}{w}$ ) satisfying $\lVert\nabla{\psi_{1}}(\bm{z})\rVert>\lVert\bm{v}\rVert^{-1/2}/2$ and $\lambda\leq\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda$ . Therefore, $\mathcal{J}_{1,u,\bm{v},\lambda}=0$ unless $\lVert\bm{v}\rVert^{-1/2}/2m<\lambda\leq 2A_{10}$ . Since $\int_{\lambda>0}d^{\times}{\lambda}\,\nu_{1}(\bm{t}/\lambda)=1$ for all $\bm{t}\in\mathbb{R}^{m}\setminus\{\bm{0}\}$ , we obtain (under (8.14)) the decomposition

(8.18)

\mathcal{J}_{u,\bm{v}}(\phi)-\mathcal{J}_{0,u,\bm{v}}=\int_{\lVert\bm{v}\rVert^{-1/2}/2m}^{2A_{10}}d^{\times}{\lambda}\,\mathcal{J}_{1,u,\bm{v},\lambda}.

Fix functions $\omega_{1},\dots,\omega_{m}\in C^{\infty}_{c}(\mathbb{R}^{m})$ , all supported on $\operatorname{Supp}{\nu_{1}}$ , such that $\omega_{1}+\dots+\omega_{m}=\nu_{1}$ and we have $1/2\leq\lvert t_{j}\rvert\leq 2m$ for all $\bm{t}\in\operatorname{Supp}{\omega_{j}}$ . Let $\mathcal{J}_{1,j,u,\bm{v},\lambda}$ denote $\mathcal{J}_{1,u,\bm{v},\lambda}$ with $\omega_{j}$ in place of $\nu_{1}$ ; clearly $\mathcal{J}_{1,u,\bm{v},\lambda}=\sum_{1\leq j\leq m}\mathcal{J}_{1,j,u,\bm{v},\lambda}$ .

We proceed based on the following rough idea: if $\lVert\nabla{\psi_{1}(\bm{z})}\rVert\gg\lVert\bm{v}\rVert^{-1/2}$ with a large constant, integration by parts over $\bm{z}$ is useful; and if $\lVert\nabla{\psi_{1}(\bm{z})}\rVert\ll\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}$ with a small constant, integration by parts over $u$ is useful (if $\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert\gg 1$ with any constant).

Let $\lambda\in[4\lVert\bm{v}\rVert^{-1/2},2A_{10}]$ . Since $\lambda\geq 4\lVert\bm{v}\rVert^{-1/2}$ , the supports of $\nu_{0}(\nabla{\psi_{1}}(\bm{z})/\lVert\bm{v}\rVert^{-1/2})$ and $\nu_{1}(\nabla{\psi_{1}}(\bm{z})/\lambda)$ are disjoint, so $\overline{\nu}_{0}(\nabla{\psi_{1}}(\bm{z})/\lVert\bm{v}\rVert^{-1/2})\omega_{j}(\nabla{\psi_{1}}(\bm{z})/\lambda)=\omega_{j}(\nabla{\psi_{1}}(\bm{z})/\lambda)$ . Thus

(8.19)

\lvert\tilde{u}\rvert^{m/2}\mathcal{J}_{1,j,u,\bm{v},\lambda}=\int_{\mathbb{R}^{m}}d\bm{z}\,w_{2,j,0}(\bm{z})e(\lVert\bm{v}\rVert\psi_{1}(\bm{z})/\lvert\tilde{u}\rvert^{1/2}),

where $w_{2,j,0}=w_{2,j,0,u,\bm{v},\lambda}(\bm{z})\colonequals\omega_{j}(\nabla{\psi_{1}}(\bm{z})/\lambda)\phi(\bm{z}/\lvert\tilde{u}\rvert^{1/2})$ .

Since $\operatorname{Supp}{\omega_{j}}\subseteq\{\bm{t}\in\mathbb{R}^{m}:1/2\leq\lvert t_{j}\rvert\leq 2m\}\cap\operatorname{Supp}{\nu_{1}}$ , we have

(8.20)

\lVert\partial_{z_{j}}{\psi_{1}}(\bm{z})\rVert\geq\lambda/2,\quad\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda,

for all $\bm{z}\in\operatorname{Supp}{w_{2,j,0}}$ . For $k\geq 1$ , recursively define $w_{2,j,k}=w_{2,j,k,u,\bm{v},\lambda}(\bm{z})$ (a sequence of smooth functions $\mathbb{R}^{m}\to\mathbb{C}$ supported on $\operatorname{Supp}{w_{2,j,0}}$ ) by setting

(8.21)

w_{2,j,k}\colonequals-(\partial_{z_{j}}{\psi_{1}})^{k}\cdot\partial_{z_{j}}(w_{2,j,k-1}/(\partial_{z_{j}}{\psi_{1}})^{k})

Integrating by parts $k$ times in (8.19), we get (cf. [heath1996new]*(5.4), from the proof of Lemma 10)

(8.22)

\lvert\tilde{u}\rvert^{m/2}\mathcal{J}_{1,j,u,\bm{v},\lambda}=\int_{\mathbb{R}^{m}}d\bm{z}\,\frac{w_{2,j,k,u,\bm{v},\lambda}(\bm{z})}{(\lVert\bm{v}\rVert\partial_{z_{j}}{\psi_{1}}(\bm{z})/\lvert\tilde{u}\rvert^{1/2})^{k}}\cdot e(\lVert\bm{v}\rVert\psi_{1}(\bm{z})/\lvert\tilde{u}\rvert^{1/2}).

We claim that for each $k\geq 0$ , there exists a smooth function $w_{3,j,k}\colon\mathbb{R}^{m(k+2)+3}\to\mathbb{C}$ of the form $w_{3,j,k}=w_{3,j,k}(\bm{a}_{1,0},\dots,\bm{a}_{1,k},\bm{a}_{2,0},a_{2,1},a_{3},a_{4})$ , defined in terms of $F$ , $j$ , $k$ (independently of $u$ , $\bm{v}$ , $\lambda$ ), such that for all $u$ , $\bm{v}$ , $\lambda$ currently under consideration (namely, satisfying $\lVert\bm{v}\rVert\geq 1$ , (8.14), and $\lambda\in[4\lVert\bm{v}\rVert^{-1/2},2A_{10}]$ ), and for all $\bm{z}\in\operatorname{Supp}{w_{2,j,0}}$ , we have

(8.23)

\lambda^{k}\cdot w_{2,j,k}(\bm{z})=w_{3,j,k}{\left(\frac{\nabla{\psi_{1}}(\bm{z})}{\lambda},\nabla{\partial_{z_{j}}^{1}{\psi_{1}}}(\bm{z}),\dots,\nabla{\partial_{z_{j}}^{k}{\psi_{1}}}(\bm{z}),\frac{\bm{z}}{\lvert\tilde{u}\rvert^{1/2}},\frac{1}{\lvert\tilde{u}\rvert^{1/2}},\frac{\lambda}{\partial_{z_{j}}{\psi_{1}}},\lambda\right)}.

This claim can be easily proven by induction on $k\geq 0$ , where for $k=0$ we take

w_{3,j,0}(\bm{a}_{1,0},\bm{a}_{2,0},a_{2,1},a_{3},a_{4})\colonequals\omega_{j}(\bm{a}_{1,0})\phi(\bm{a}_{2,0}),

and for $k\geq 1$ we use the product and chain rules in (8.21) (to compute $\lambda^{k}w_{2,j,k}=\lambda\cdot\lambda^{k-1}w_{2,j,k}$ in terms of $\lambda^{k-1}w_{2,j,k-1}$ and its $z_{j}$ -derivatives), and observe that (for $l\geq 1$ )

	$\displaystyle\lambda\cdot\partial_{z_{j}}{\left(\frac{\psi_{1}}{\lambda}\right)}=\partial_{z_{j}}^{1}{\psi_{1}},\quad\lambda\cdot\partial_{z_{j}}(\partial_{z_{j}}^{l})=\lambda\cdot\partial_{z_{j}}^{l+1},\quad\lambda\cdot\partial_{z_{j}}(\bm{z})=\lambda,$
	$\displaystyle\lambda\cdot\partial_{z_{j}}{\left(\frac{\lambda}{\partial_{z_{j}}{\psi_{1}}}\right)}=-(\partial_{z_{j}}^{2}{\psi_{1}})\cdot\left(\frac{\lambda}{\partial_{z_{j}}{\psi_{1}}}\right)^{2},\quad(\partial_{z_{j}}{\psi_{1}})^{k}\cdot\partial_{z_{j}}{\left(\frac{\lambda}{(\partial_{z_{j}}{\psi_{1}})^{k}}\right)}=(\partial_{z_{j}}^{2}{\psi_{1}})\cdot\frac{-k\lambda}{\partial_{z_{j}}{\psi_{1}}}.$

(This proof uses the smoothness of $\psi_{1}$ , and the nonvanishing of $\partial_{z_{j}}{\psi_{1}}$ on $\operatorname{Supp}{w_{2,j,0}}$ .)

By (8.9), we have $\partial^{\bm{\alpha}}{\psi_{1}}(\bm{z})\ll_{\bm{\alpha}}1$ whenever $\lvert\bm{\alpha}\rvert\geq 2$ and $\bm{z}\in\mathscr{W}$ . Inserting this and the bounds (8.20), (8.14), (8.9), and $\lambda\ll 1$ into (8.23), we find that $\lambda^{k}w_{2,j,k}(\bm{z})\ll_{w_{3,j,k}}1$ for all $\bm{z}\in\mathbb{R}^{m}$ . Thus (8.22) and (8.20) immediately give (for $\lambda\in[4\lVert\bm{v}\rVert^{-1/2},2A_{10}]$ )

(8.24)

\lvert\tilde{u}\rvert^{m/2}\mathcal{J}_{1,j,u,\bm{v},\lambda}\ll_{k}\int_{\bm{z}\in\mathscr{W}}d\bm{z}\,\frac{\bm{1}_{\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda}}{(\lambda^{2}\lVert\bm{v}\rVert/\lvert\tilde{u}\rvert^{1/2})^{k}},

under (8.14). Our derivation of (8.24) has essentially followed the proof of Lemma 8.6 (with a small but important twist: we use $\operatorname{Supp}{w_{2,j,k}}\subseteq\operatorname{Supp}{w_{2,j,0}}$ to get the factor $\bm{1}_{\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda}$ ), but in some key ranges we need to go further. We need to decide when to integrate by parts over $\tilde{u}\in\mathbb{R}$ ; this will be informed by the next two paragraphs.

By (8.8) and the inverse function theorem, we know that for each $\bm{z}\in\mathscr{W}$ , there exists an open neighborhood $U\subseteq\mathbb{R}^{m}$ of $\bm{z}$ such that the gradient map $\nabla{F}\colon\mathbb{R}^{m}\to\mathbb{R}^{m}$ maps $U$ diffeomorphically onto $\nabla{F}(U)$ . But $\nabla{F}(\mathscr{W})$ is compact (since $\mathscr{W}$ is compact), so there exists a constant $\eta_{6}=\eta_{6}(F,w)>0$ such that for any $\bm{z}\in\mathscr{W}$ and $\bm{b}\in[-1,1]^{m}$ with $\lVert\nabla{F}(\bm{z})-\bm{b}\rVert\leq\eta_{6}$ , there exists $\bm{s}\in(\nabla{F})^{-1}(\bm{b})$ with $\lVert\bm{z}-\bm{s}\rVert\leq\eta_{6}^{-1}\lVert\nabla{F}(\bm{z})-\bm{b}\rVert$ .

We apply this as follows. Let $\bm{b}=\operatorname{sgn}(u)\tilde{\bm{v}}$ and $\bm{z}\in\mathscr{W}$ , and suppose $\lVert\nabla{\psi_{1}(\bm{z})}\rVert\leq\eta_{6}$ . Then there exists $\bm{s}\in\mathbb{R}^{m}$ with $\nabla{\psi_{1}}(\bm{s})=\bm{0}$ and $\lVert\bm{z}-\bm{s}\rVert\leq\eta_{6}^{-1}\lVert\nabla{\psi_{1}(\bm{z})}\rVert$ . Since $\nabla{\psi_{1}}(\bm{s})=\bm{0}$ , Taylor expansion of $\psi_{1}$ at $\bm{s}$ gives $\psi_{1}(\bm{z})=\psi_{1}(\bm{s})+O_{F,w}(\lVert\bm{z}-\bm{s}\rVert^{2})$ . Furthermore, $\nabla{\psi_{1}}(\bm{s})=\bm{0}$ implies $\bm{b}\cdot\bm{s}=\nabla{F}(\bm{s})\cdot\bm{s}=3F(\bm{s})$ , so $\lvert\psi_{1}(\bm{s})\rvert=\lvert 2F(\bm{s})\rvert$ . But $\Delta(\bm{b})=\Delta(\nabla{F}(\bm{s}))\ll_{F,w}\lvert F(\bm{s})\rvert$ by (8.1). Combining the above, we get

(8.25)

\lvert\psi_{1}(\bm{z})\rvert\geq\eta_{7}\lvert\Delta(\tilde{\bm{v}})\rvert-\eta_{7}^{-1}\lVert\nabla{\psi_{1}(\bm{z})}\rVert^{2}\quad\textnormal{(for some $\eta_{7}=\eta_{7}(F,w)>0$)}.

Let $A_{11}\colonequals 1+\sup_{\bm{a}\in[-1,1]^{m}}{\lvert\Delta(\bm{a})\rvert}$ , and let $\eta_{8}\colonequals m^{-1}\min(\eta_{6}/A_{11}^{1/2},\eta_{7}/2)$ .

Our remaining analysis breaks into two cases: $\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert\geq(4/\eta_{8})^{2}$ and $\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert<(4/\eta_{8})^{2}$ .

Suppose first that $\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert<(4/\eta_{8})^{2}$ . Consider a $\tilde{u}\in\mathbb{R}$ satisfying (8.14). By (8.15), the right-hand side of (8.24) is $\ll_{k}\lambda^{m}(\lambda^{2}\lVert\bm{v}\rVert)^{-k}=\lVert\bm{v}\rVert^{-m/2}(\lambda\lVert\bm{v}\rVert^{1/2})^{m-2k}$ . Choosing $k=\lceil m/2+1\rceil$ and inserting (8.24) into (8.18), we then get

\mathcal{J}_{u,\bm{v}}(\phi)-\mathcal{J}_{0,u,\bm{v}}-\int_{\lVert\bm{v}\rVert^{-1/2}/2m}^{4\lVert\bm{v}\rVert^{-1/2}}d^{\times}{\lambda}\,\mathcal{J}_{1,u,\bm{v},\lambda}\ll_{m}\lVert\bm{v}\rVert^{-m/2},

since $\int_{4\lVert\bm{v}\rVert^{-1/2}}^{\infty}d^{\times}{\lambda}\,(\lambda\lVert\bm{v}\rVert^{1/2})^{m-2k}=\int_{4}^{\infty}d^{\times}{a}\,a^{m-2k}\ll 1$ (via the substitution $a=\lambda\lVert\bm{v}\rVert^{1/2}$ ). But if we simply take absolute values in $\mathcal{J}_{0,u,\bm{v}}$ (see (8.16)) and in $\mathcal{J}_{1,u,\bm{v},\lambda}$ (see (8.17)), and then apply (8.15), we get $\mathcal{J}_{0,u,\bm{v}}\ll\lVert\bm{v}\rVert^{-m/2}$ and $\mathcal{J}_{1,u,\bm{v},\lambda}\ll\lVert\bm{v}\rVert^{-m/2}$ for $\lambda\in[\lVert\bm{v}\rVert^{-1/2}/2m,4\lVert\bm{v}\rVert^{-1/2}]$ . Integrating over $\lambda$ in the previous display, we conclude that $\mathcal{J}_{u,\bm{v}}(\phi)\ll_{m}\lVert\bm{v}\rVert^{-m/2}$ (under (8.14)). Hence by (8.13) (after writing $u=\lVert\bm{v}\rVert\tilde{u}$ ) we have

\mathscr{J}_{r,\bm{v}}(q,\phi)\ll_{b}(1+\lVert\bm{v}\rVert)^{-b}+\lVert\bm{v}\rVert\int_{1/3A_{9}\leq\lvert\tilde{u}\rvert\leq 3A_{9}}d\tilde{u}\,\lvert q(r\lVert\bm{v}\rVert\tilde{u})\rvert\cdot\lVert\bm{v}\rVert^{-m/2}.

But $q(r\lVert\bm{v}\rVert\tilde{u})\ll_{b}(1+r\lVert\bm{v}\rVert)^{-b}$ (under (8.14)). Thus $\mathscr{J}_{r,\bm{v}}(q,\phi)\ll_{b}\lVert\bm{v}\rVert^{1-m/2}(1+r\lVert\bm{v}\rVert)^{-b}$ , which suffices for Proposition 8.4 (since $\lVert\bm{v}\rVert\geq 1$ and $\lVert\Delta(\bm{v}/M)\bm{v}\rVert\leq\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert\ll 1$ ).

For the rest of the proof, assume $\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert\geq(4/\eta_{8})^{2}$ , so that $4\lVert\bm{v}\rVert^{-1/2}\leq\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}$ . We will integrate by parts over $\tilde{u}$ in the integrals

\mathscr{J}_{0}\colonequals\int_{\mathbb{R}}d\tilde{u}\,q(r\lVert\bm{v}\rVert\tilde{u})w_{1}(\tilde{u})\mathcal{J}_{0,u,\bm{v}},\quad\mathscr{J}_{1,\lambda}\colonequals\int_{\mathbb{R}}d\tilde{u}\,q(r\lVert\bm{v}\rVert\tilde{u})w_{1}(\tilde{u})\mathcal{J}_{1,u,\bm{v},\lambda}

for $\lambda\leq\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}$ . It is crucial to work in terms of $\bm{z}$ rather than $\bm{x}=\bm{z}/\lvert\tilde{u}\rvert^{1/2}$ . Before proceeding, note that inserting (8.18) into (8.13) (and writing $u=\lVert\bm{v}\rVert\tilde{u}$ ) gives

(8.26)

\mathscr{J}_{r,\bm{v}}-\lVert\bm{v}\rVert\cdot\mathscr{J}_{0}-\lVert\bm{v}\rVert\cdot\int_{\lVert\bm{v}\rVert^{-1/2}/2m}^{2A_{10}}d^{\times}{\lambda}\,\mathscr{J}_{1,\lambda}\ll_{b}(1+\lVert\bm{v}\rVert)^{-b}.

Let $\bm{z}\in\mathscr{W}$ , and suppose $\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}$ . By the definition of $\eta_{8}$ , we then have $\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq\eta_{6}$ and $\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq\eta_{7}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}/2$ , so by (8.25), we have

(8.27)

\lvert\psi_{1}(\bm{z})\rvert\geq 3\eta_{7}\lvert\Delta(\tilde{\bm{v}})\rvert/4.

For convenience, let $\mathscr{W}(\lambda)\colonequals\{\bm{z}\in\mathscr{W}:\lVert\nabla{\psi_{1}}(\bm{z})\rVert\leq m\lambda\}$ for each $\lambda>0$ .

We first bound $\mathscr{J}_{0}$ . For each $\tilde{u}\in\operatorname{Supp}{w_{1}}$ , the condition (8.14) holds, so if we plug in the definition of $\mathcal{J}_{0,u,\bm{v}}$ (see (8.16)), and then switch the order of $u$ , $\bm{z}$ , we may rewrite $\mathscr{J}_{0}$ as

\int_{\mathscr{W}(\lVert\bm{v}\rVert^{-1/2})}d\bm{z}\,\nu_{0}{\left(\frac{\nabla{\psi_{1}}(\bm{z})}{\lVert\bm{v}\rVert^{-1/2}}\right)}\int_{\mathbb{R}}\frac{d\tilde{u}}{\lvert\tilde{u}\rvert^{m/2}}\,q(r\lVert\bm{v}\rVert\tilde{u})w_{1}(\tilde{u})\phi{\left(\frac{\bm{z}}{\lvert\tilde{u}\rvert^{1/2}}\right)}e{\left(\frac{\lVert\bm{v}\rVert\psi_{1}(\bm{z})}{\lvert\tilde{u}\rvert^{1/2}}\right)}.

Here $\psi_{1}$ is independent of $\lvert\tilde{u}\rvert$ (when $\operatorname{sgn}(u)$ is fixed). Therefore, for each $\bm{z}\in\mathscr{W}(\lVert\bm{v}\rVert^{-1/2})$ , the inner integral (over $\tilde{u}$ ) is $\ll_{k}(1+r\lVert\bm{v}\rVert)^{-k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{-k}$ by (8.27) and Lemma 8.6, because $\lvert\tilde{u}\rvert\asymp 1$ for $\tilde{u}\in\operatorname{Supp}{w_{1}}$ and we have $\partial_{\tilde{u}}^{l}q(r\lVert\bm{v}\rVert\tilde{u})=(r\lVert\bm{v}\rVert)^{l}q^{(l)}(r\lVert\bm{v}\rVert\tilde{u})\ll_{l,k}(1+r\lVert\bm{v}\rVert)^{-k}$ for all $l,k\geq 0$ (since $q$ is Schwartz). Applying this inner integral estimate for each $\bm{z}\in\mathscr{W}(\lVert\bm{v}\rVert^{-1/2})$ , and then using (8.15), we get $\mathscr{J}_{0}\ll_{k}\lVert\bm{v}\rVert^{-m/2}(1+r\lVert\bm{v}\rVert)^{-k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{-k}$ .

One can similarly prove $\mathscr{J}_{1,\lambda}\ll_{k}\lVert\bm{v}\rVert^{-m/2}(1+r\lVert\bm{v}\rVert)^{-k}$ for $\lambda\in[\lVert\bm{v}\rVert^{-1/2}/2m,4\lVert\bm{v}\rVert^{-1/2}]$ .

Now suppose $4\lVert\bm{v}\rVert^{-1/2}\leq\lambda\leq\min(\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2},2A_{10})$ . Let $\mathscr{J}_{1,j,\lambda}$ denote $\mathscr{J}_{1,\lambda}$ with $\mathcal{J}_{1,j,u,\bm{v},\lambda}$ in place of $\mathcal{J}_{1,u,\bm{v},\lambda}$ . Then $\mathscr{J}_{1,\lambda}=\sum_{1\leq j\leq m}\mathscr{J}_{1,j,\lambda}$ . By (8.22), we have (for all integers $l\geq 0$ )

\mathscr{J}_{1,j,\lambda}=\int_{\bm{z}\in\mathscr{W}(\lambda)}d\bm{z}\int_{\mathbb{R}}\frac{d\tilde{u}}{\lvert\tilde{u}\rvert^{m/2}}\,q(r\lVert\bm{v}\rVert\tilde{u})w_{1}(\tilde{u})\frac{w_{2,j,l,u,\bm{v},\lambda}(\bm{z})e(\lVert\bm{v}\rVert\psi_{1}(\bm{z})/\lvert\tilde{u}\rvert^{1/2})}{(\lVert\bm{v}\rVert\partial_{z_{j}}{\psi_{1}}(\bm{z})/\lvert\tilde{u}\rvert^{1/2})^{l}}.

Since $\psi_{1}$ is independent of $\lvert\tilde{u}\rvert$ , and we have $\lVert\bm{z}\rVert\ll 1$ and $\lvert\tilde{u}\rvert\asymp 1$ for all $(\bm{z},\tilde{u})\in\mathscr{W}\times\operatorname{Supp}{w_{1}}$ , we find by (8.20), (8.23), (8.27), and Lemma 8.6 (applied to the inner integral over $\tilde{u}$ , for each $\bm{z}\in\mathscr{W}(\lambda)$ for which there exists $u\in\mathbb{R}$ with $w_{1}(\tilde{u})w_{2,j,l,u,\bm{v},\lambda}(\bm{z})\neq 0$ ) that

\mathscr{J}_{1,j,\lambda}\ll_{k,l}\int_{\bm{z}\in\mathscr{W}(\lambda)}d\bm{z}\,\frac{(1+r\lVert\bm{v}\rVert)^{-k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{-k}}{(\lambda^{2}\lVert\bm{v}\rVert)^{l}}\ll\frac{\lambda^{m}(1+r\lVert\bm{v}\rVert)^{-k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{-k}}{(\lambda^{2}\lVert\bm{v}\rVert)^{l}}

for all integers $k,l\geq 0$ , where in the final step we use (8.15).

Finally, suppose $\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}\leq\lambda\leq 2A_{10}$ . Then (8.24) and (8.15) directly give (for $k,l\geq 0$ )

\mathscr{J}_{1,\lambda}\ll_{l}\int_{\mathbb{R}}d\tilde{u}\,\frac{\lvert q(r\lVert\bm{v}\rVert\tilde{u})w_{1}(\tilde{u})\rvert\cdot\lambda^{m}}{(\lambda^{2}\lVert\bm{v}\rVert)^{l}}\ll_{k}\frac{(1+r\lVert\bm{v}\rVert)^{-k}\lVert\bm{v}\rVert^{-m/2}}{(\lambda\lVert\bm{v}\rVert^{1/2})^{2l-m}}.

Inserting our work from the last four paragraphs (ignoring the last one if $\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}>2A_{10}$ ) into the left-hand side of (8.26), and applying (8.26) with $b=\lceil m/2+2k\rceil$ , we get the bound

\frac{\mathscr{J}_{r,\bm{v}}}{\lVert\bm{v}\rVert}\ll_{k,l}\frac{1+\int_{\lambda\geq 4\lVert\bm{v}\rVert^{-1/2}}d^{\times}{\lambda}\,(\lambda\lVert\bm{v}\rVert^{1/2})^{m-2l}}{\lVert\bm{v}\rVert^{m/2}(1+r\lVert\bm{v}\rVert)^{k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{k}}+\frac{\int_{\lambda\geq\eta_{8}\lvert\Delta(\tilde{\bm{v}})\rvert^{1/2}}d^{\times}{\lambda}\,(\lambda\lVert\bm{v}\rVert^{1/2})^{m-2l+2k}}{\lVert\bm{v}\rVert^{m/2}(1+r\lVert\bm{v}\rVert)^{k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{k}}.

Taking $l=\lceil m/2+k+1\rceil$ , and evaluating both integrals over $\lambda$ (the first being $\ll 1$ since $4\lVert\bm{v}\rVert^{-1/2}\cdot\lVert\bm{v}\rVert^{1/2}=4\gg 1$ , and the second being $\ll 1$ since $\eta_{8}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{1/2}\geq 4\gg 1$ ), we get $\mathscr{J}_{r,\bm{v}}\ll_{k}\lVert\bm{v}\rVert^{1-m/2}(1+r\lVert\bm{v}\rVert)^{-k}\lVert\Delta(\tilde{\bm{v}})\bm{v}\rVert^{-k}$ , which suffices for Proposition 8.4. ∎

9. New bounds on bad exponential sums

We first provide some new, general, vanishing and boundedness criteria for $S_{\bm{c}}(p^{l})$ , which when combined with classical estimates from [hooley1986HasseWeil, heath1998circle] will (under Conjecture 1.5) allow us to break a critical $\epsilon$ -barrier behind (2.17). Recall $S_{\bm{c}}(n)$ , $S^{\natural}_{\bm{c}}(n)$ from (2.8), (2.9).

Lemma 9.1.

Let $\bm{c}\in\mathcal{S}_{1}$ , and let $p$ be a prime.

(1)

If $p\nmid\gcd(\Delta(\bm{c}),\partial_{c_{1}}{\Delta}(\bm{c}),\dots,\partial_{c_{m}}{\Delta}(\bm{c}))$ , then $S^{\natural}_{\bm{c}}(p)\ll_{m}1$ .
(2)

If $v_{p}(\Delta(\bm{c}))\leq 1$ , then $S^{\natural}_{\bm{c}}(p^{2})\ll_{F}1$ .
(3)

We have $S_{\bm{c}}(p^{l})=0$ for all integers $l\geq 2+v_{p}(\Delta(\bm{c}))$ .

Proof.

(1): If $p\mid 6$ , then $\lvert S_{\bm{c}}(p)\rvert\leq p^{1+m}$ trivially, so $\lvert S^{\natural}_{\bm{c}}(p)\rvert\leq p^{(1+m)/2}\ll_{m}1$ . Now suppose $p\nmid 6\gcd(\Delta(\bm{c}),\partial_{c_{1}}{\Delta}(\bm{c}),\dots,\partial_{c_{m}}{\Delta}(\bm{c}))$ . Then by (2.1), we have

p\nmid\operatorname{disc}(F),\quad p\nmid 6,\quad p\nmid\gcd(\operatorname{disc}(F,\bm{c}),\partial_{c_{1}}{\operatorname{disc}(F,\bm{c})},\dots,\partial_{c_{m}}{\operatorname{disc}(F,\bm{c})}).

Therefore, by (2.12) and [wang2023dichotomous]*Theorem 1.1, and Proposition 2.7(2) $\Rightarrow$ (1) with $(n,r,d)=(m-1,2,3)$ , we have $\lvert E^{\natural}_{\bm{c}}(p)\rvert\leq 72\cdot 9^{m}$ . Yet $E^{\natural}_{F}(p)\ll_{m}1$ , by (2.12) and the Weil conjectures (since $p\nmid\operatorname{disc}(F)$ ). Plugging these two estimates into (2.13), we get $S^{\natural}_{\bm{c}}(p)\ll_{m}1$ .

(If $m=6$ and $F$ is diagonal, one can also improve (1) to a “codimension-three” statement, by using [wang2023dichotomous]*Theorem 1.3 in place of [wang2023dichotomous]*Theorem 1.1.)

We now turn to (2)–(3). For any vector $\bm{u}\in\mathbb{Z}^{m}$ , let $v_{p}(\bm{u})\colonequals v_{p}(\gcd(u_{1},\dots,u_{m}))$ . Write $\bm{c}=p^{g}\tilde{\bm{c}}$ , where $g=v_{p}(\bm{c})<\infty$ and $\tilde{\bm{c}}\in\mathcal{S}_{1}$ . For integers $u,d\geq 0$ , let

\mathscr{B}^{(d)}(\bm{c};p,u)\colonequals\bigcup_{\lambda\in\mathbb{Z}:\,p\nmid\lambda}\{\bm{x}\in\mathbb{Z}^{m}:p\nmid\bm{x},\;p^{u}\mid F(\bm{x}),\;p^{u}\mid\tilde{\bm{c}}\cdot\bm{x},\;p^{d}\mid\nabla{F}(\bm{x})-\lambda\bm{c}\}.

For any $u,d\geq 0$ and $\bm{x}\in\mathscr{B}^{(d)}(\bm{c};p,u)$ , Corollary 2.3 (with $\tilde{\bm{c}}$ in place of $\bm{c}$ ) implies

(9.1)

\gcd(p^{u},p^{2d})\mid\Delta(\tilde{\bm{c}}).

For any set $A\subseteq\mathbb{Z}^{m}$ that can be written as a finite union of residue classes $\mathcal{R}\subseteq\mathbb{Z}^{m}$ , let $\mu(A)$ be the density of $A$ in $\mathbb{Z}^{m}$ . Now let $l\geq 2+2g$ be an integer, and let $d=\lfloor l/2\rfloor\geq 1+g$ ; then by [wang2023_isolating_special_solutions]*Proposition 7.4, we have

(9.2)

p^{-lm}\varphi(p^{l})S^{\prime}_{\bm{c}}(p^{l})=p^{2l+g}\mu(\mathscr{B}^{(d)}(\bm{c};p,l))-p^{2l-2+g}\mu(\mathscr{B}^{(d)}(\bm{c};p,l-1)),

where $S^{\prime}_{\bm{c}}(p^{l})$ (the “restriction to $p\nmid\bm{x}$ ” of $S_{\bm{c}}(p^{l})$ ) is defined as in [wang2023_isolating_special_solutions]*(7.5).

(2): Suppose $v_{p}(\Delta(\bm{c}))\leq 1$ . Then $p\nmid\bm{c}$ (since $\deg{\Delta}\geq 2$ ), so $g=0$ and $\tilde{\bm{c}}=\bm{c}$ . In particular, $S_{\bm{c}}(p^{2})=S^{\prime}_{\bm{c}}(p^{2})$ by [wang2023_isolating_special_solutions]*Lemma 7.2. By (9.2) (with $l=2$ and $d=\lfloor l/2\rfloor=1$ ), it remains to analyze $\mathscr{B}^{(1)}(\bm{c};p,u)$ for $u\in\{1,2\}$ . By (9.1), we have $\mathscr{B}^{(1)}(\bm{c};p,2)=\emptyset$ (since $p^{2}\nmid\Delta(\tilde{\bm{c}})$ ). We now analyze $\mathscr{B}^{(1)}(\bm{c};p,1)$ .

For each $\bm{a}\in\mathscr{B}^{(1)}(\bm{c};p,1)$ , the variety $F(\bm{x})=\bm{c}\cdot\bm{x}=0$ in $\mathbb{P}^{m-1}_{\mathbb{F}_{p}}$ is singular at the point $[a_{1}:\cdots:a_{m}]\in\mathbb{P}^{m-1}(\mathbb{F}_{p})$ . Since $p\nmid\bm{c}$ , this variety is isomorphic to a cubic hypersurface $f(y_{1},\dots,y_{m-1})=0$ in $\mathbb{P}^{m-2}_{\mathbb{F}_{p}}$ , and has a singular locus of dimension $\leq 0$ if $p\nmid\operatorname{disc}(F)$ (by [wang2023dichotomous]*Theorem 2.3, due to Zak). So if $p\nmid 3\operatorname{disc}(F)$ , then $p^{m}\mu(\mathscr{B}^{(1)}(\bm{c};p,1))\leq(p-1)\cdot 2^{m-1}$ by Bézout’s theorem (applied to the system $\nabla{f}=0$ in $\mathbb{P}^{m-2}_{\mathbb{F}_{p}}$ ; note that $3f=\bm{y}\cdot\nabla{f}$ ).

But if $p\mid 3\operatorname{disc}(F)$ , then $p^{m}\mu(\mathscr{B}^{(1)}(\bm{c};p,1))\leq p^{m}-1$ trivially. So in every case, $p^{m}\mu(\mathscr{B}^{(1)}(\bm{c};p,1))\ll_{F}p-1$ . Now by (2.9) and (9.2), we have

S^{\natural}_{\bm{c}}(p^{2})=\frac{S_{\bm{c}}(p^{2})}{p^{1+m}}=\frac{S^{\prime}_{\bm{c}}(p^{2})}{p^{1+m}}=\frac{p^{m-1}}{\varphi(p^{2})}\cdot\left(p^{4}\cdot 0-p^{2}\cdot\mu(\mathscr{B}^{(1)}(\bm{c};p,1))\right)\ll_{F}\frac{p(p-1)}{\varphi(p^{2})}=1.

(3): Take a counterexample $(\bm{c},l)\in\mathcal{S}_{1}\times\mathbb{Z}_{\geq 0}$ with $l$ minimal. Then $l\geq 2+v_{p}(\Delta(\bm{c}))$ and $S_{\bm{c}}(p^{l})\neq 0$ . So $l\geq 2+2g$ , because $\deg{\Delta}\geq 2$ . Furthermore, $d=\lfloor l/2\rfloor\geq(l-1)/2$ . By (9.1), we have $\mathscr{B}^{(d)}(\bm{c};p,u)=\emptyset$ for all $u\geq l-1$ , since $2d\geq l-1$ and $p^{l-1}\nmid\Delta(\bm{c})$ . So (9.2) gives $S^{\prime}_{\bm{c}}(p^{l})=0$ , whence $S_{\bm{c}}(p^{l})\neq S^{\prime}_{\bm{c}}(p^{l})$ . If $g=0$ , this contradicts [wang2023_isolating_special_solutions]*Lemma 7.2. If $g=1$ , then $l\geq 4$ , so $S_{\bm{c}}(p^{l})-S^{\prime}_{\bm{c}}(p^{l})=0$ by [wang2023_isolating_special_solutions]*Lemma 7.2(2); again, a contradiction. Now suppose $g\geq 2$ ; then $p^{2}\mid\bm{c}$ and $l-3\geq 2+v_{p}(\Delta(\bm{c}/p^{2}))$ (since $\deg{\Delta}\geq 3/2$ ), so $S_{\bm{c}/p^{2}}(p^{l-3})=0$ by the minimality hypothesis. But [wang2023_isolating_special_solutions]*Lemma 7.2(2) then gives $S_{\bm{c}}(p^{l})-S^{\prime}_{\bm{c}}(p^{l})=0$ ; another contradiction. Therefore, no counterexample $(\bm{c},l)\in\mathcal{S}_{1}\times\mathbb{Z}_{\geq 0}$ to (3) in fact exists.

For an alternative, more algorithmic and computational approach to (2)–(3) (at least when $F$ is diagonal), see [wang2022thesis]*§7.2 and [wang2021_HLH_vs_RMT]*Appendix D. ∎

Remark 9.2.

If $F$ were quadratic (rather than cubic), then Lemma 9.1(1) would be false whenever $2\mid m$ . See [wang2022thesis]*Remark 7.2.4 or [wang2023dichotomous]*sentence after Theorem 1.1.

Let $\mathcal{N}_{\leq}(t)\colonequals\{n\geq 1:p\mid n\Rightarrow v_{p}(n)\leq t\}$ and $\mathcal{N}_{\geq}(t)\colonequals\{n\geq 1:p\mid n\Rightarrow v_{p}(n)\geq t\}$ for each integer $t\geq 1$ . For any integers $N,t\geq 1$ , we have (see e.g. [bateman1958theorem])

(9.3)

\lvert\{N\leq n<2N:n\in\mathcal{N}_{\geq}(t)\}\rvert\ll_{t}N^{1/t}.

Recall $\mathcal{N}_{\bm{c}}$ from (2.2). Lemma 9.1(1) has a useful unconditional consequence:

Proposition 9.3.

Let $A\in\mathbb{R}_{\geq 1}$ . Uniformly over reals $Z,N>0$ with $N\leq Z^{3}$ , we have

(9.4)

\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\biggl{(}\,\sum_{n\in\mathcal{N}_{\bm{c}}\cap\mathcal{N}_{\leq}(1)\cap[N,2N)}n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\biggr{)}^{\!2A}\ll_{F,A,\epsilon}Z^{m}\min(N,Z)^{-1+\epsilon}.

Proof.

The case $N<1$ is trivial, so assume $N\geq 1$ . Let $\mathcal{D}=\{1,2,4,8,\ldots\}$ . Let

(9.5)

\mathcal{N}_{\bm{c},0}\colonequals\{n\in\mathcal{N}_{\leq}(1):p\mid n\Rightarrow p\mid\gcd(\Delta(\bm{c}),\partial_{c_{1}}{\Delta}(\bm{c}),\dots,\partial_{c_{m}}{\Delta}(\bm{c}))\}

for each $\bm{c}\in\mathbb{Z}^{m}$ . For each $Q\in\mathcal{D}$ , let $\mathcal{S}_{3}(Q)$ be the set of tuples $\bm{c}\in\mathcal{S}_{1}$ for which $\mathcal{N}_{\bm{c},0}\cap[Q,2Q)\neq\emptyset$ . Note that $1\in\mathcal{N}_{\bm{c},0}$ , so $\mathcal{S}_{3}(1)=\mathcal{S}_{1}$ .

It is known (e.g. by [heath1983cubic]*Lemma 11) that $S^{\natural}_{\bm{c}}(p)\ll_{m}p^{1/2}$ for all $\bm{c}\in\mathbb{Z}^{m}$ and primes $p$ . So for all $\bm{c}\in\mathcal{S}_{1}$ and $n\in\mathcal{N}_{\leq}(1)$ , Lemma 9.1(1) implies

S^{\natural}_{\bm{c}}(n)\ll_{m,\epsilon}n^{\epsilon}\cdot\max_{q\mid n:\,q\in\mathcal{N}_{\bm{c},0}}{q^{1/2}}\ll n^{\epsilon}\cdot\left(\max{\{Q\in\mathcal{D}\cap[1,n]:\bm{c}\in\mathcal{S}_{3}(Q)\}}\right)^{1/2}.

Therefore, the left-hand side of (9.4) is at most $O_{m,A,\epsilon}(1)$ times the quantity

(9.6)

\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}:\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert^{2A}\sum_{\begin{subarray}{c}Q\in\mathcal{D}\cap[1,2N):\\ \bm{c}\in\mathcal{S}_{3}(Q)\end{subarray}}\frac{N^{\epsilon}Q^{A}}{N^{A}}=\sum_{Q\in\mathcal{D}\cap[1,2N)}\frac{N^{\epsilon}Q^{A}}{N^{A}}\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{3}(Q):\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert^{2A}.

Let $\epsilon\in(0,1)$ , and consider an individual $Q\in\mathcal{D}$ . Lemma 6.17, and Hölder over $\bm{c}$ , give

(9.7)

\sum_{\bm{c}\in\mathcal{S}_{3}(Q)\cap[-Z,Z]^{m}}\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert^{2A}\ll_{A,\epsilon}\lvert\mathcal{S}_{3}(Q)\cap[-Z,Z]^{m}\rvert^{1-\epsilon}\cdot(Z^{m}N^{\epsilon})^{\epsilon}.

On the other hand, the scheme $\Delta=\partial_{c_{1}}{\Delta}=\dots=\partial_{c_{m}}{\Delta}=0$ in $\mathbb{A}^{m}_{\mathbb{Q}}$ has dimension $\leq m-2$ (since $\Delta=0$ is generically smooth, due to the absolute irreducibility of $\Delta$ ). So by a quantitative form of [ekedahl1991infinite]’s geometric sieve (see [bhargava2014geometric]*Theorem 3.3 for the case of prime moduli, which extends to square-free moduli as in [bhargava2021galois]*§5, Case III), we have

(9.8)

\lvert\mathcal{S}_{3}(Q)\cap[-Z,Z]^{m}\rvert\ll_{\Delta,\epsilon}Z^{m}Q^{-1+\epsilon}+Z^{m-1+\epsilon};

this bound follows from Lang–Weil if $Q\leq Z$ (since $\sum_{n\in\mathcal{N}_{\leq}(1)\cap[Q,2Q)}O_{\epsilon}(\frac{Z^{m}}{n^{2-\epsilon}})\ll_{\epsilon}Z^{m}Q^{-1+\epsilon}$ ; cf. [bhargava2014geometric]*(16)), and from elimination theory if $Q>Z$ (cf. [bhargava2014geometric]*(17) and [bhargava2021galois]*the three paragraphs after Proposition 33).

Upon inserting (9.8) into (9.7), we find that the right-hand side of (9.6) is

\ll_{A,\epsilon}\sum_{Q\in\mathcal{D}\cap[1,2N)}\frac{N^{\epsilon}Q^{A}}{N^{A}}\cdot Z^{m}N^{\epsilon^{2}}(Q^{-(1-\epsilon)^{2}}+Z^{-(1-\epsilon)^{2}})\ll_{A,\epsilon}N^{2\epsilon}\cdot Z^{m}(N^{-(1-\epsilon)^{2}}+Z^{-(1-\epsilon)^{2}}),

since $A\geq 1$ . Since $N^{\epsilon}\leq\min(N,Z)^{3\epsilon}$ , the bound (9.4) follows immediately. ∎

We now build on Conjecture 1.5.

Conjecture 9.4 (SFSC_q,3).

There exists a real $\eta_{9}=\eta_{9}(\Delta)>0$ such that

(9.9)

\#\{\bm{c}\in[-Z,Z]^{m}:\exists\;\textnormal{$q\in\mathcal{N}_{\geq}(2)\cap[Q,2Q)$ with $q\mid\Delta(\bm{c})$}\}\ll_{\Delta}Z^{m}Q^{-\eta_{9}}

holds uniformly over reals $Z,Q\geq 1$ with $Q\leq Z^{3}$ .

Proposition 9.5.

Assume Conjecture 1.5. Then Conjecture 9.4 holds.

Proof.

Suppose $Z,Q\geq 1$ are reals with $Q\leq Z^{3}$ .

Case 1: $Q\leq Z$ . Recall the notation $N(P;q)$ from §4. Let $\delta=(\deg{\Delta})^{-1}$ and $\epsilon\in(0,\delta)$ . The left-hand side of (9.9) is at most

(9.10)

\sum_{q\in\mathcal{N}_{\geq}(2)\cap[Q,2Q)}N(\Delta;q)\cdot O{\left(\frac{Z^{m}}{q^{m}}\right)}\ll_{\epsilon}\frac{Z^{m}}{Q^{\delta/2-\epsilon}}\sum_{q\in\mathcal{N}_{\geq}(2)}q^{\delta/2-\epsilon}\cdot\frac{N(\Delta;q)}{q^{m}}.

Let $p$ be a prime and $l\geq 2$ an integer. By (4.3), we have $N(\Delta;p^{l})\ll_{\Delta}p^{l(m-\delta)}$ . On the other hand, by Hensel’s lemma (over the smooth locus of $\Delta=0$ ) and Lang–Weil (over the singular locus of $\Delta=0$ ), we have $N(\Delta;p^{2})\ll_{\Delta}p^{2m-2}$ , and thus $N(\Delta;p^{l})\leq p^{(l-2)m}N(\Delta;p^{2})\ll_{\Delta}p^{lm-2}$ . So by the Chinese remainder theorem, the right-hand side of (9.10) is

\leq\frac{Z^{m}}{Q^{\delta/2-\epsilon}}\prod_{p}\biggl{(}1+\sum_{l\geq 2}p^{(\delta/2-\epsilon)l}\cdot O_{\Delta}(p^{-\max(\delta l,2)})\biggr{)}\leq\frac{Z^{m}}{Q^{\delta/2-\epsilon}}\biggl{(}1+\sum_{l\geq 2}O_{\Delta}(p^{-1-\epsilon l})\biggr{)}

(since $(\delta/2-\epsilon)l-\max(\delta l,2)\leq-1-\epsilon l$ ). Since $\epsilon>0$ , it follows that the right-hand side of (9.10) is $\ll_{\Delta,\epsilon}Z^{m}Q^{-\delta/2+\epsilon}$ . Hence the left-hand side of (9.9) is $\ll_{\Delta,\epsilon}Z^{m}Q^{-\delta/2+\epsilon}$ .

Case 2: $Q>Z$ . Suppose $q\in\mathcal{N}_{\geq}(2)\cap[Q,2Q)$ . Let $s$ be the largest square divisor of $q$ ; then $s\geq q^{2/3}\geq Q^{2/3}>Z^{2/3}$ . The integer $s^{1/2}$ thus lies in $(Z^{1/4},2Q^{1/2})$ , and hence either has a prime factor $p>Z^{1/4}$ (so that $p\in[P,2P)$ for some $P\in[Z^{1/4},Q^{1/2}]$ ), or an integer factor $d\in(Z^{1/4},Z^{1/2}]$ (so that $d^{2}\in[D,2D)$ for some $D\in[Z^{1/2},Z]$ ). So by Conjecture 1.5 (with $P\in[Z^{1/4},Q^{1/2}]$ ) and Case 1 (with $D\in[Z^{1/2},Z]$ in place of $Q$ ), the left-hand side of (9.9) is

\ll_{\Delta,\epsilon}Z^{m}(Z^{1/4})^{-\eta_{0}}+Z^{m}(Z^{1/2})^{-\delta/2+\epsilon}.

Since $Z\geq Q^{1/3}$ , it follows that (9.9) holds with $\eta_{9}=\frac{1}{12}\min(\eta_{0},\frac{9}{10}(\deg{\Delta})^{-1})$ . ∎

For all $W\in\mathbb{Z}[\bm{c}]$ and $Z,N,A\in\mathbb{R}_{>0}$ , let

(9.11)

\Sigma_{14}^{W,A}(Z,N)\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}:\,W(\bm{c})\neq 0}\biggl{(}\,\sum_{n\in\mathcal{N}_{\bm{c}}\cap[N,2N)}n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\biggr{)}^{\!A}.

Close analogs of the moment $\Sigma_{14}^{1,A}(Z,N)$ have been considered for $A=1$ classically (e.g. in [heath1983cubic, hooley1986HasseWeil, heath1998circle]), and for $A=2$ in [wang2023_large_sieve_diagonal_cubic_forms]*Propositions 4.12 and A.1.

Conjecture 9.6 (B2).

Let $Z,N\in\mathbb{R}_{\geq 1}$ with $N\leq Z^{3}$ . Then $\Sigma_{14}^{1,2}(Z,N)\ll_{F,\epsilon}Z^{m+\epsilon}$ .

Proposition 9.7.

Suppose $F$ is diagonal. Then Conjecture 9.6 holds. Also, for $W=c_{1}\cdots c_{m}$ and $1\leq N\leq Z^{3}$ , we have $\Sigma_{14}^{1,2}(Z,N)-\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m-1+\epsilon}N^{1/3}$ .

Proposition 9.7 is essentially due to [wang2023_large_sieve_diagonal_cubic_forms]; but we need to go one step further.

Conjecture 9.8 (B3G).

Let $1\leq A\leq 2$ . There exists a nonzero polynomial $W=W_{F,A}\in\mathbb{Z}[\bm{c}]$ , and a real $\eta_{10}=\eta_{10}(F,A)>0$ , such that if $1\leq N\leq Z^{3}$ , then $\Sigma_{14}^{W,A}(Z,N)\ll_{F,A}Z^{m}N^{-\eta_{10}}$ .

One can extend Conjecture 9.8 to $A=2+\delta$ , almost for free (see Proposition 10.3), but it is not clear to us what the limit is. Also, for $1\leq A\leq 2$ , one might hope for $\eta_{10}(F,A)\approx A/2$ to be admissible, by comparing with Proposition 9.3 or [sarnak1991bounds]*Conjecture 1.

Proposition 9.9.

Suppose $F$ is diagonal. Assume Conjecture 1.5. Then Conjecture 9.8 holds with $W=c_{1}\cdots c_{m}$ (for all $A\in[1,2]$ ).

Remark 9.10.

If for each $m^{\prime}\in\{2,3,\dots,m\}$ , one assumed an analog of Conjecture 1.5 with $m^{\prime}$ in place of $m$ , then one could prove Proposition 9.9 with $W=1$ . See [wang2022thesis]*Conjecture 7.3.7 (B3) and Remark 7.3.13 and [wang2021_HLH_vs_RMT]*Lemma 7.43 for details.

We need a classical pointwise bound on $S^{\natural}_{\bm{c}}(n)$ . For integers $c\neq 0$ , let $\operatorname{sq}(c)\colonequals\prod_{p^{2}\mid c}p^{v_{p}(c)}$ and $\operatorname{cub}(c)\colonequals\prod_{p^{3}\mid c}p^{v_{p}(c)}$ . Also let $\operatorname{sq}(0)\colonequals 0$ and $\operatorname{cub}(0)\colonequals 0$ .

Proposition 9.11 ([hooley1986HasseWeil, heath1998circle]; see e.g. [wang2023_large_sieve_diagonal_cubic_forms]*Proposition 4.9).

Assume $F$ is diagonal. Let $\epsilon>0$ . There exists $A_{12}(F,\epsilon)\in\mathbb{R}_{\geq 1}$ such that if $\bm{c}\in\mathbb{Z}^{m}$ and $n\in\mathbb{Z}_{\geq 1}$ , then

n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\leq A_{12}(F,\epsilon)n^{\epsilon}\prod_{1\leq j\leq m}{\gcd}{\left(\operatorname{cub}(n)^{2},\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{j}))^{3}\right)}^{1/12}.

One could replace the $n^{\epsilon}$ in Proposition 9.11 with $O(1)^{\omega(n)}$ . This would be important if one wanted to try to prove a softer version of Proposition 9.9 (conditional on a softer version of Conjecture 1.5). We do not explore this direction in the present paper, since it would seem to involve complicated divisor-type sums with square-full parts and discriminant divisibility conditions (which might require complicated parameterizations to handle).

Proposition 9.11 is an explicit stratification result for $S_{\bm{c}}(n)$ , based on $v_{p}(c_{j})$ for $p\mid n$ . It would be very nice to have a usable (perhaps less explicit) replacement for Proposition 9.11 when $F$ is no longer diagonal. Work such as [denef1984rationality, pas1989uniform] could conceivably help.

We now prove Propositions 9.7 and 9.9. For convenience, let $\mathcal{S}_{4}\colonequals\{\bm{c}\in\mathcal{S}_{1}:c_{1}\cdots c_{m}\neq 0\}$ .

Proof of Proposition 9.7.

Let $W=c_{1}\cdots c_{m}$ . Suppose $Z,N\in\mathbb{R}_{\geq 1}$ with $N\leq Z^{3}$ . For each $\bm{c}\in\mathcal{S}_{4}$ , Proposition 9.11 implies $n^{-1/2}S^{\sharp}_{\bm{c}}(n)\ll_{F,\epsilon}n^{\epsilon}\prod_{1\leq j\leq m}\operatorname{sq}(c_{j})^{1/4}$ . Plugging this into $\Sigma_{14}^{W,2}(Z,N)$ (see (9.11)), and using Lemma 2.1 to bound $\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert$ , we get $\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{\epsilon}\prod_{1\leq j\leq m}\sum_{1\leq\lvert c\rvert\leq Z}\operatorname{sq}(c_{j})^{1/2}$ . However, by (9.3), we have

(9.12)

\sum_{1\leq\lvert c\rvert\leq Z}\operatorname{sq}(c)^{1/2}\ll\sum_{e\in\mathcal{N}_{\geq}(2)}e^{1/2}\cdot(Z/e)\ll Z\log(1+Z).

Therefore, $\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m+\epsilon}$ . The statement $\Sigma_{14}^{1,2}(Z,N)-\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m-1+\epsilon}N^{1/3}$ holds by a similar calculation with Proposition 9.11, Lemma 2.1, and (9.12), ending with

\operatorname{cub}(n)^{(m-t)/3}\prod_{1\leq i\leq t}\sum_{1\leq\lvert c_{i}\rvert\leq Z}\operatorname{sq}(c_{i})^{1/2}\ll_{t,\epsilon}N^{(m-t)/3}Z^{t+\epsilon}\leq N^{1/3}Z^{m-1+\epsilon}

for $n\in[N,2N)$ , $t\in[1,m-1]$ ; cf. [wang2023_large_sieve_diagonal_cubic_forms]*(4.11) in the proof of Proposition 4.12. Since $N\leq Z^{3}$ , we obtain $\Sigma_{14}^{1,2}(Z,N)\ll_{F,\epsilon}Z^{m+\epsilon}$ , proving Conjecture 9.6. ∎

Proof of Proposition 9.9.

Suppose $1\leq N\leq Z^{3}$ . For each $\bm{c}\in\mathbb{Z}^{m}$ and $n\in\mathbb{Z}_{\geq 1}$ , let

S^{\sharp}_{\bm{c}}(n)\colonequals\prod_{p\mid n}\max\left(1,\frac{\lvert S^{\natural}_{\bm{c}}(p^{v_{p}(n)})\rvert}{p^{v_{p}(n)/2}}\right).

Let $P\geq 1$ be a real parameter to be specified later. Let

\Sigma_{15}\colonequals\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{1}:\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}\,\biggl{(}\,\sum_{\begin{subarray}{c}n\in\mathcal{N}_{\bm{c}}\cap[N,2N):\\ S^{\sharp}_{\bm{c}}(n)<P\end{subarray}}\frac{\lvert S^{\natural}_{\bm{c}}(n)\rvert}{n^{1/2}}\biggr{)}^{\!2},\quad\Sigma_{16}\colonequals\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{4}:\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}\,\biggl{(}\,\sum_{\begin{subarray}{c}n\in\mathcal{N}_{\bm{c}}\cap[N,2N):\\ S^{\sharp}_{\bm{c}}(n)\geq P\end{subarray}}\frac{\lvert S^{\natural}_{\bm{c}}(n)\rvert}{n^{1/2}}\biggr{)}^{\!2}.

Let $W=c_{1}\cdots c_{m}$ ; then $\Sigma_{14}^{W,2}(Z,N)\leq 2(\Sigma_{15}+\Sigma_{16})$ , since $(a+b)^{2}\leq 2(a^{2}+b^{2})$ for $a,b\in\mathbb{R}$ . We will bound $\Sigma_{15}$ conditionally (using Conjecture 1.5), and $\Sigma_{16}$ unconditionally (using the diagonality of $F$ ). We will lose factors of $Z^{\epsilon}$ at first, and then remove $Z^{\epsilon}$ later.

We first handle $\Sigma_{15}$ . Given $\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}$ , let

\begin{split}\mathcal{N}_{\bm{c},1}&\colonequals\{n\in\mathcal{N}_{\geq}(2):p\mid n\Rightarrow\operatorname{lcm}(p^{2},p^{v_{p}(n)-1})\nmid\Delta(\bm{c})\},\\ \mathcal{N}_{\bm{c},2}&\colonequals\{n\in\mathcal{N}_{\geq}(2):p\mid n\Rightarrow\operatorname{lcm}(p^{2},p^{v_{p}(n)-1})\mid\Delta(\bm{c})\}.\end{split}

Any integer $n\geq 1$ can be written uniquely as $qn_{1}n_{2}$ , where $q$ , $n_{1}$ , $n_{2}$ are pairwise coprime integers satisfying $q\in\mathcal{N}_{\leq}(1)$ and $n_{j}\in\mathcal{N}_{\bm{c},j}$ (for $j\in\{1,2\}$ ). We then have

n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\leq\min_{d\mid n}(S^{\sharp}_{\bm{c}}(n/d)\cdot d^{-1/2}\lvert S^{\natural}_{\bm{c}}(d)\rvert)\leq S^{\sharp}_{\bm{c}}(n)\min(1,n_{1}^{-1/2}\lvert S^{\natural}_{\bm{c}}(n_{1})\rvert,q^{-1/2}\lvert S^{\natural}_{\bm{c}}(q)\rvert)

by the definition of $S^{\sharp}$ . Here $S^{\natural}_{\bm{c}}(n_{1})\ll_{F}1$ by Lemma 9.1(2)–(3). Also, the integer $\prod_{p\mid n_{2}}\operatorname{lcm}(p^{2},p^{v_{p}(n_{2})-1})\in\mathcal{N}_{\geq 2}\cap[n_{2}^{2/3},n_{2}]$ divides $\Delta(\bm{c})$ . Since $\lvert\mathcal{N}_{\bm{c}}\cap[1,2N)\rvert\ll_{\epsilon}(ZN)^{\epsilon}$ (by Lemma 2.1) and $\min(q,n_{1},n_{2})\geq n^{1/3}$ , we conclude that

\Sigma_{15}\ll_{F,\epsilon}P^{2}Z^{\epsilon}\sum_{\bm{c}\in\mathcal{S}_{1}\cap[-Z,Z]^{m}}\biggl{(}\,\bm{1}_{\bm{c}\in\mathcal{S}_{5}}+\frac{1}{(N^{1/3})^{1/2}}+\sum_{q\in\mathcal{N}_{\bm{c}}\cap\mathcal{N}_{\leq}(1)\cap[N^{1/3},2N)}\frac{\lvert S^{\natural}_{\bm{c}}(q)\rvert}{q^{1/2}}\biggr{)}^{\!2},

where $\mathcal{S}_{5}$ denotes the set of tuples $\bm{c}\in\mathcal{S}_{1}$ for which there exists $d\in\mathcal{N}_{\geq 2}\cap[(N^{1/3})^{2/3},2N)$ with $d\mid\Delta(\bm{c})$ . Since $(r+s+t)^{2}\leq 3(r^{2}+s^{2}+t^{2})$ (for $r,s,t\in\mathbb{R}$ ), the sum over $\bm{c}$ on the right above is $\ll_{F,\epsilon}Z^{m}(N^{2/9})^{-\eta_{9}}+Z^{m}N^{-1/3}+Z^{m}N^{\epsilon}\min(N^{1/3},Z)^{-1+\epsilon}$ , by Propositions 9.5 and 9.3. Since $N\leq Z^{3}$ , it follows that

(9.13)

\Sigma_{15}\ll_{F,\epsilon}P^{2}Z^{m+\epsilon}(N^{-2\eta_{9}/9}+N^{-1/3}).

We next handle $\Sigma_{16}$ , by introducing a new Ekedahl-type idea. For each $\bm{c}\in\mathcal{S}_{1}$ , let

\mathcal{N}_{\bm{c},3}(P)\colonequals\{n\in\mathcal{N}_{\bm{c}}:S^{\natural}_{\bm{c}}(n)\neq 0,\;S^{\sharp}_{\bm{c}}(n)\geq P\}.

Let $\epsilon\in(0,1)$ (to be specified); suppose $P^{1/10}\geq A_{12}(F,\epsilon)\cdot(2N)^{\epsilon}$ . Now consider an individual $\bm{c}\in\mathcal{S}_{4}\cap[-Z,Z]^{m}$ and $n\in\mathcal{N}_{\bm{c},3}(P)\cap[N,2N)$ . Here $S^{\natural}_{\bm{c}}(n)\neq 0$ , so $\prod_{p\mid\operatorname{cub}(n)}p^{v_{p}(n)-1}\mid\Delta(\bm{c})$ by Lemma 9.1(3). And $S^{\sharp}_{\bm{c}}(n)\geq P$ , so

P^{9/10}\leq S^{\sharp}_{\bm{c}}(n)/P^{1/10}\leq S^{\sharp}_{\bm{c}}(n)/A_{12}(F,\epsilon)n^{\epsilon}\leq\prod_{1\leq j\leq m}\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{j}))^{1/4},

by Proposition 9.11. Thus there exists $j$ with $d\colonequals\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{j}))\geq P^{3.6/m}$ . Clearly $d\in\mathcal{N}_{\geq}(2)$ . Also, $d\mid c_{j}$ , so $d\leq Z$ (since $W(\bm{c})\neq 0$ implies $c_{j}\neq 0$ ); and $d\mid\operatorname{cub}(n)\mid\prod_{p\mid\operatorname{cub}(n)}p^{2(v_{p}(n)-1)}\mid\Delta(\bm{c})^{2}$ . Letting $\mathcal{S}_{6}(j,d)\colonequals\{\bm{c}\in\mathcal{S}_{4}:d\mid\gcd(c_{j},\Delta(\bm{c})^{2})\}$ , it follows (by taking $n\in\mathcal{N}_{\bm{c},3}(P)\cap[N,2N)$ with $n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert$ maximal, if such an $n$ exists) that

\Sigma_{16}\leq\sum_{1\leq j\leq m}\sum_{\begin{subarray}{c}d\in\mathcal{N}_{\geq}(2):\\ P^{3.6/m}\leq d\leq Z\end{subarray}}\,\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{6}(j,d):\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}\,\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert^{2}\cdot d^{1/2}\prod_{\begin{subarray}{c}1\leq i\leq m:\\ i\neq j\end{subarray}}\operatorname{sq}(c_{i})^{1/2}.

Using Lemma 2.1 to bound $\lvert\mathcal{N}_{\bm{c}}\cap[N,2N)\rvert$ , we get

(9.14)

\Sigma_{16}\ll_{F,\epsilon}Z^{\epsilon}\sum_{1\leq j\leq m}\sum_{\begin{subarray}{c}d\in\mathcal{N}_{\geq}(2):\\ P^{3.6/m}\leq d\leq Z\end{subarray}}\,\sum_{\begin{subarray}{c}\bm{c}\in\mathcal{S}_{6}(j,d):\\ \lVert\bm{c}\rVert\leq Z\end{subarray}}d^{1/2}\prod_{\begin{subarray}{c}1\leq i\leq m:\\ i\neq j\end{subarray}}\operatorname{sq}(c_{i})^{1/2}.

For notational simplicity, suppose $j=m$ . For $k\in\{0,1\}$ , let

\mathcal{S}_{6,k}(m)\colonequals\{(c_{1},\dots,c_{m-1})\in(\mathbb{Z}\setminus\{0\})^{m-1}:\bm{1}_{\Delta(c_{1},\dots,c_{m-1},0)\neq 0}=k\}.

For each $(c_{1},\dots,c_{m-1})\in\mathcal{S}_{6,1}(m)\cap[-Z,Z]^{m-1}$ , we have

\sum_{\begin{subarray}{c}d\in\mathcal{N}_{\geq}(2):\\ P^{3.6/m}\leq d\leq Z\end{subarray}}\,\sum_{\begin{subarray}{c}1\leq\lvert c_{m}\rvert\leq Z:\\ (c_{1},\dots,c_{m})\in\mathcal{S}_{6}(m,d)\end{subarray}}d^{1/2}\leq\sum_{\begin{subarray}{c}d\mid\Delta(c_{1},\dots,c_{m-1},0)^{2}:\\ P^{3.6/m}\leq d\leq Z\end{subarray}}\,\sum_{\begin{subarray}{c}1\leq\lvert c_{m}\rvert\leq Z:\\ d\mid c_{m}\end{subarray}}d^{1/2}\ll_{F,\epsilon}Z^{\epsilon}\cdot Z/(P^{3.6/m})^{1/2},

by the divisor bound for $\Delta(c_{1},\dots,c_{m-1},0)^{2}\neq 0$ . Therefore, the contribution to the right-hand side of (9.14) from $\bm{c}$ with $(c_{1},\dots,c_{m-1})\in\mathcal{S}_{6,1}(m)$ (or the analogous condition if $j\neq m$ ) is

\ll_{F,\epsilon}\frac{Z^{1+2\epsilon}}{P^{1.8/m}}\prod_{1\leq i\leq m-1}\sum_{1\leq\lvert c_{i}\rvert\leq Z}\operatorname{sq}(c_{i})^{1/2}\ll_{m,\epsilon}\frac{Z^{m+3\epsilon}}{P^{1.8/m}},

where we use (9.12) in the final step.

On the other hand, for any $c_{1},\dots,c_{m-2}\in\mathbb{Z}$ , there are at most $\deg{\Delta}$ integers $c_{m-1}$ for which $(c_{1},\dots,c_{m-1})\in\mathcal{S}_{6,0}(m)$ . (This is because for diagonal $F$ , the $c_{i}^{\deg{\Delta}}$ coefficient of $\Delta$ is nonzero for all $i$ ; see e.g. [heath1998circle]*(4.2).) Therefore, the contribution to the right-hand side of (9.14) from $\bm{c}$ with $(c_{1},\dots,c_{m-1})\in\mathcal{S}_{6,0}(m)$ (or the analogous condition if $j\neq m$ ) is

\ll_{F,\epsilon}Z^{\epsilon}\sum_{\begin{subarray}{c}d\in\mathcal{N}_{\geq}(2):\\ P^{3.6/m}\leq d\leq Z\end{subarray}}\,\sum_{\begin{subarray}{c}1\leq\lvert c_{m}\rvert\leq Z:\\ d\mid c_{m}\end{subarray}}d^{1/2}\max_{1\leq\lvert c_{m-1}\rvert\leq Z}{\operatorname{sq}(c_{m-1})^{1/2}}\prod_{1\leq i\leq m-2}\sum_{1\leq\lvert c_{i}\rvert\leq Z}\operatorname{sq}(c_{i})^{1/2},

and thus (by (9.12)) $\ll_{F,\epsilon}Z^{2\epsilon}\cdot Z\cdot Z^{1/2}\cdot Z^{m-2}=Z^{m-1/2+2\epsilon}$ .

The previous two paragraphs imply that the right-hand side of (9.14) is $\ll_{F,\epsilon}Z^{m+3\epsilon}(P^{-1.8/m}+Z^{-1/2})$ . This, combined with (9.13), gives

\Sigma_{14}^{W,2}(Z,N)\leq 2(\Sigma_{15}+\Sigma_{16})\ll_{F,\epsilon}P^{2}Z^{m+\epsilon}(N^{-2\eta_{9}/9}+N^{-1/3})+Z^{m+3\epsilon}(P^{-1.8/m}+Z^{-1/2}).

Suppose $\epsilon\leq 2\min(1,\eta_{9})/270$ , and let $P=A_{12}(F,\epsilon)^{10}2^{10\epsilon}N^{2\min(1,\eta_{9})/27}\geq 1$ ; then we get

\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m+3\epsilon}(N^{-3.6\min(1,\eta_{9})/27m}+Z^{-1/2})\ll Z^{m+3\epsilon}N^{-\min(1,\eta_{9})/9m},

since $m\geq 4$ and $Z\geq N^{1/3}$ . It follows (upon redefining $\epsilon$ , now that we can forget about $P$ ) that $\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m+\epsilon}N^{-\min(1,\eta_{9})/9m}$ for all $\epsilon>0$ .

It remains to replace $Z^{\epsilon}$ with $N^{\epsilon}$ . This is trivial unless $N$ is very small relative to $Z$ . Expanding the square in (9.11), and noting that $S_{\bm{c}}(n)$ depends only on $\bm{c}\bmod{n}$ , gives

\Sigma_{14}^{W,2}(Z,N)\leq\sum_{n_{1},n_{2}\in[N,2N)}\left(1+\frac{2Z}{n_{1}n_{2}}\right)^{m}\sum_{\begin{subarray}{c}1\leq\bm{a}\leq n_{1}n_{2}:\\ \operatorname{rad}(n_{1}n_{2})\mid\Delta(\bm{a})\end{subarray}}(n_{1}n_{2})^{-1/2}\lvert S^{\natural}_{\bm{a}}(n_{1})S^{\natural}_{\bm{a}}(n_{2})\rvert

for all $Z,N\geq 1$ . Yet for any $n_{1},n_{2}\geq 1$ and $\bm{a}\in[1,n_{1}n_{2}]^{m}$ , there exists $\bm{c}\in\mathcal{S}_{4}\cap[1,Bn_{1}n_{2}]^{m}$ with $\bm{c}\equiv\bm{a}\bmod{n_{1}n_{2}}$ , where $B=1+\deg(c_{1}\cdots c_{m}\Delta(\bm{c}))$ . So

\Sigma_{14}^{W,2}(Z,N)\geq\sum_{n_{1},n_{2}\in[N,2N)}\sum_{\begin{subarray}{c}1\leq\bm{a}\leq n_{1}n_{2}:\\ \operatorname{rad}(n_{1}n_{2})\mid\Delta(\bm{a})\end{subarray}}(n_{1}n_{2})^{-1/2}\lvert S^{\natural}_{\bm{a}}(n_{1})S^{\natural}_{\bm{a}}(n_{2})\rvert

for all $Z\geq 4BN^{2}$ . Hence for all $Z\geq N^{2}$ we have

\Sigma_{14}^{W,2}(Z,N)\leq(3Z/N^{2})^{m}\cdot\Sigma_{14}^{W,2}(4BN^{2},N)\ll_{F,\epsilon}(Z/N^{2})^{m}(4BN^{2})^{m+\epsilon}N^{-\min(1,\eta_{9})/9m},

since $1\leq N\leq(4BN^{2})^{3}$ . For all $Z\geq N^{1/3}$ (including $Z<N^{2}$ , where $Z^{\epsilon}<N^{2\epsilon}$ ), then,

\Sigma_{14}^{W,2}(Z,N)\ll_{F,\epsilon}Z^{m}N^{\epsilon-\min(1,\eta_{9})/9m}.

Therefore, Conjecture 9.8 holds with $W=c_{1}\cdots c_{m}$ and $\eta_{10}(F,2r)=r\min(1,\eta_{9})/10m$ (for $r=1$ at first, and then for $r\in[1/2,1]$ by Hölder over $\bm{c}$ ). ∎

Remark 9.12.

Handling $\operatorname{sq}(c_{i})^{1/2}$ takes care, because $\sum_{1\leq\lvert c\rvert\leq Z}\operatorname{sq}(c)^{1/2}$ remains roughly the same size even if we restrict $\lvert c\rvert$ to $\mathcal{N}_{\geq}(2)$ (a rather sparse set). One could take $W=1$ in Proposition 9.9 if we were only interested in $\Sigma_{14}^{W,A}(Z,N)$ for $A\in[1,2)$ , or for $N\leq Z^{3-\delta}$ .

To end §9, we prove a result clarifying the nature of Conjecture 1.5 for diagonal $F$ :

Proposition 9.13.

Assume Conjecture 1.5 holds when $(m,F)=(4,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})$ . Then Conjecture 1.5 holds whenever $F$ is diagonal and $m\geq 4$ .

Proof.

(This result is not used in the rest of the paper, so we confine ourselves to a sketch.)

Assume $1\leq P\leq Z^{3/2}$ . Recall $\mathcal{N}_{\bm{c},0}$ from (9.5). By [bhargava2014geometric]*Theorem 3.3, we have

(9.15)

\#\{\bm{c}\in[-Z,Z]^{m}:\exists\;\textnormal{$p\in[P,2P)\cap\mathcal{N}_{\bm{c},0}$}\}\ll_{\Delta}Z^{m}P^{-\Theta(1)}.

By (9.15), we see that for any given $F$ , Conjecture 1.5 is equivalent to the statement

(9.16)

\#\{\bm{c}\in[-Z,Z]^{m}:\exists\;\textnormal{$p\in[P,2P)\setminus\mathcal{N}_{\bm{c},0}$ with $p^{2}\mid\Delta(\bm{c})$}\}\ll Z^{m}P^{-\Theta(1)}.

Let $\mathcal{U}_{1},\mathcal{U}_{2}\subseteq\mathbb{P}^{m-1}_{\mathbb{Z}}$ be the smooth loci of the hypersurfaces $F=0$ , $\Delta=0$ , respectively, over $\mathbb{Z}$ . The gradient $\nabla{F}$ defines a Gauss map $[\nabla{F}]\colon\mathcal{U}_{1}\to\mathbb{P}^{m-1}_{\mathbb{Z}}$ ; let $\mathcal{U}_{3}\subseteq\mathcal{U}_{1}$ be the inverse image of $\mathcal{U}_{2}$ under this map. The map $[\nabla{F}]\colon\mathcal{U}_{3}\to\mathcal{U}_{2}$ is an isomorphism over $\mathbb{Q}$ (by the biduality theorem), and thus an isomorphism over $\mathbb{Z}$ (since $\mathcal{U}_{2}$ , $\mathcal{U}_{3}$ are flat over $\mathbb{Z}$ ). In particular, $\mathcal{U}_{2}(\mathbb{Z}/p^{2}\mathbb{Z})=[\nabla{F}](\mathcal{U}_{3}(\mathbb{Z}/p^{2}\mathbb{Z}))$ . Furthermore, it is known that $\mathcal{U}_{3}$ lies in the open subscheme $\det(\operatorname{Hess}{F}(\bm{x}))\neq 0$ of $\mathbb{P}^{m-1}_{\mathbb{Z}}$ . These two facts, plus (9.15), imply that (9.16) is equivalent to

(9.17)

\begin{split}\#\{\bm{c}\in[-Z,Z]^{m}:\exists\;&p\in[P,2P),\;\bm{x}\in\mathbb{Z}^{m},\;\lambda\in[1,p-1]\textnormal{ with }\\ &p\nmid\det(\operatorname{Hess}{F}(\bm{x})),\;p^{2}\mid F(\bm{x}),\nabla{F}(\bm{x})-\lambda\bm{c}\}\ll Z^{m}P^{-\Theta(1)}.\end{split}

We would like to convert each “exists” into a sum. Each $p$ on the left-hand side of (9.17) satisfies $p\mid\Delta(\bm{c})$ , so there are at most finitely many possibilities for $p$ if $\bm{c}\in\mathcal{S}_{1}$ . Also, $\lvert\mathcal{S}_{0}\cap[-Z,Z]^{m}\rvert\cdot P\ll_{m}Z^{m-1/2}$ by (2.4). Therefore, the statement (9.17) is equivalent to

(9.18)

\sum_{\bm{c}\in[-Z,Z]^{m}}\sum_{p\in[P,2P)}\sum_{1\leq\bm{x}\leq p^{2}}\bm{1}_{p\nmid\det(\operatorname{Hess}{F}(\bm{x}))}\bm{1}_{p^{2}\mid F(\bm{x})}\mathbb{E}_{1\leq\lambda\leq p-1}[\bm{1}_{p^{2}\mid\nabla{F}(\bm{x})-\lambda\bm{c}}]\ll Z^{m}P^{-\Theta(1)}.

Now suppose $F=F_{1}x_{1}^{3}+\dots+F_{m}x_{m}^{3}$ (where $F_{j}\in\mathbb{Z}\setminus\{0\}$ ), and for each $G,\lambda\in\mathbb{Z}$ let

T_{3/2,G,\lambda}(a,p^{2})\colonequals\sum_{\begin{subarray}{c}c\in[-Z,Z],\;1\leq x\leq p^{2}:\\ p\nmid x,\;p^{2}\mid 3Gx^{2}-\lambda c\end{subarray}}e_{p^{2}}(aGx^{3})\in\mathbb{R}.

Then the left-hand side of (9.18) equals $1/(p-1)$ times

(9.19)

\sum_{\begin{subarray}{c}p\in[P,2P),\\ 1\leq a\leq p^{2},\;1\leq\lambda\leq p-1\end{subarray}}\prod_{1\leq j\leq m}T_{3/2,F_{j},\lambda}(a,p^{2})\ll_{F}Z^{m-4}\sum_{1\leq j\leq 4}\sum_{\begin{subarray}{c}p\in[P,2P),\\ 1\leq a\leq p^{2},\;1\leq\lambda\leq p-1\end{subarray}}T_{3/2,F_{j},\lambda}(a,p^{2})^{4},

since $T_{3/2,G,\lambda}(a,p^{2})\ll_{G}Z$ trivially (by considering the cases $p\nmid 3G$ and $p\mid 3G$ separately) and $\prod_{1\leq j\leq 4}\lvert T_{3/2,F_{j},\lambda}(a,p^{2})\rvert\leq\sum_{1\leq j\leq 4}T_{3/2,F_{j},\lambda}(a,p^{2})^{4}$ (since $T_{3/2,G,\lambda}\in\mathbb{R}$ ).

Since Conjecture 1.5 holds by assumption when $F=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}$ , it also holds when $F=F_{j}\cdot(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})$ for any fixed $j$ (since scaling $\Delta$ does not affect the truth of Conjecture 1.5). So by (9.19), the statement (9.18) holds for all diagonal $F$ (assuming $m\geq 4$ ). Hence (9.17), (9.16), and Conjecture 1.5 hold too. ∎

10. Delta endgame

Throughout §10, assume $2\mid m$ . Recall $\Sigma(X,\mathcal{S})$ , $\Sigma^{\natural}(X,\mathcal{S})$ from (2.15). Explicitly, we have

(10.1)

\Sigma^{\natural}(X,\mathcal{S})=\sum_{n\geq 1}\sum_{\bm{c}\in\mathcal{S}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n)=\sum_{\bm{c}\in\mathcal{S}}\sum_{n\geq 1}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n)

(by Proposition 2.4 and Fubini). We are finally prepared to analyze these sums for $\mathcal{S}\subseteq\mathcal{S}_{1}$ .

10.1. Delta decomposition

Consider an individual $\bm{c}\in\mathcal{S}_{1}$ . In view of (7.1), (7.2) and the factorization $\Phi=\Phi^{\operatorname{G}}\Phi^{\operatorname{B}}$ , we may use Lemma 5.2 with $k=2$ ,

a(\bm{n})=a(n_{0},n_{1})=(n_{0}^{-it_{0}}S^{\natural}_{\bm{c}}(n_{0})\bm{1}_{n_{0}\in\mathcal{N}_{\bm{c}}})\cdot(n_{1}^{-it_{1}}S^{\natural}_{\bm{c}}(n_{1})\bm{1}_{n_{1}\in\mathcal{N}^{\bm{c}}}),

and $f(\bm{r})=f(r_{0},r_{1})=(r_{0}r_{1})^{(1-m)/2}J_{\bm{c},X}(r_{0}r_{1})$ , to write

(10.2)

\sum_{n\geq 1}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n)=(2\pi)^{-2}\int_{\bm{N}\geq 1/2}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{2}}d\bm{t}\,g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t}),

where $\bm{N}=(N_{0},N_{1})$ runs over $[1/2,\infty)^{2}$ , where $\bm{t}=(t_{0},t_{1})\in\mathbb{R}^{2}$ , and where

(10.3)		$\displaystyle g_{\bm{c},X,\bm{N}}(\bm{r})$	$\displaystyle\colonequals(r_{0}r_{1})^{(1-m)/2}J_{\bm{c},X}(r_{0}r_{1})\cdot\nu_{2}(r_{0}/N_{0})\nu_{2}(r_{1}/N_{1}),$
(10.4)		$\displaystyle\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})$	$\displaystyle\colonequals\sum_{n_{0}\in\mathcal{N}_{\bm{c}}}\nu_{2}(n_{0}/N_{0})n_{0}^{-it_{0}}S^{\natural}_{\bm{c}}(n_{0}),\quad\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\colonequals\sum_{n_{1}\in\mathcal{N}^{\bm{c}}}\nu_{2}(n_{1}/N_{1})n_{1}^{-it_{1}}S^{\natural}_{\bm{c}}(n_{1}).$

Since $J_{\bm{c},X}(r)$ is supported on $r\leq A_{0}Y$ (by Proposition 2.4) and $\nu_{2}$ is supported on $[1,2]$ , we have $g_{\bm{c},X,\bm{N}}(\bm{r})=0$ unless $r_{0}r_{1}\leq A_{0}Y$ and $N_{j}\leq r_{j}\leq 2N_{j}$ for all $j$ . Thus $g_{\bm{c},X,\bm{N}}=0$ identically unless $N_{0}N_{1}\leq A_{0}Y$ . So $g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})=0$ unless $\bm{N}$ lies in the set

(10.5)

\mathscr{R}_{17}(X)\colonequals\{\bm{N}\geq 1/2:N_{0}N_{1}\leq A_{0}Y\}

(cf. the region $\mathscr{R}_{10}$ from (7.14)). So (10.2) holds even if we restrict $\bm{N}$ to $\mathscr{R}_{17}(X)$ .

Given $\bm{N}=(N_{0},N_{1})$ , let $N\colonequals N_{0}N_{1}$ . For integers $C\geq 1$ and reals $\lambda>0$ , let

\mathcal{S}_{1}(C,\lambda)\colonequals\{\bm{c}\in\mathcal{S}_{1}:\lVert\bm{c}\rVert\in[C,2C),\;\lvert\Delta(\bm{c}/C)\rvert\in(\lambda/2,\lambda]\}.

Assume (2.6). The definition (10.3) and Propositions 5.1 and 8.1 imply, for $\bm{c}\in\mathcal{S}_{1}(C,\lambda)$ and $b\in\mathbb{Z}_{\geq 0}$ (and multi-indices $\bm{\alpha}\geq 0$ ), that

(10.6)

N^{1/2}\partial_{\bm{c}}^{\bm{\alpha}}{g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})}\ll_{F,w,\nu_{2},b}\frac{(X/N)^{\lvert\bm{\alpha}\rvert}(N+XC)^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}(1+C/X^{1/2})^{b}(1+XC\lambda/N)^{b}}.

For each $\bm{c}\in\mathcal{S}_{1}$ , we have $\bm{c}\in\mathbb{Z}^{m}$ and $\Delta(\bm{c})\neq 0$ , so $\lVert\bm{c}\rVert\geq 1$ and $1\leq\lvert\Delta(\bm{c})\rvert\ll\lVert\bm{c}\rVert^{\deg{\Delta}}$ . Let $\mathcal{D}=\{1,2,4,8,\ldots\}$ and $\mathcal{D}_{2}=\mathcal{D}\cup\{1/d:d\in\mathcal{D}\}$ ; then we deduce that

(10.7)

\mathcal{S}_{1}\subseteq\bigcup_{C\in\mathcal{D},\;\lambda\in\mathcal{D}_{2}:\,1/C^{\deg{\Delta}}\leq\lambda\leq A_{13}}\mathcal{S}_{1}(C,\lambda)

for some $A_{13}=A_{13}(F)\geq 1$ . Also, the volume bound (4.4) implies (for all $C\geq 1$ and $\lambda\leq A_{13}$ )

(10.8)

\lvert\mathcal{S}_{1}(C,\lambda)\rvert\ll_{F}C^{m}(\lambda+C^{-1})^{1/\deg{\Delta}},

because $\lvert\Delta(\bm{z})\rvert\ll_{F}\lvert\Delta(\bm{y})\rvert+C^{-1}$ for all $\bm{y}\in[-2,2]^{m}$ and $\bm{z}\in\bm{y}+[-C^{-1},C^{-1}]^{m}$ .

10.2. Sharp delta bounds

In [wang2023_large_sieve_diagonal_cubic_forms], we used Cauchy–Schwarz on $1/L$ , $\Phi L$ over $\bm{c}$ to conditionally prove (2.17) (for diagonal $F$ with $2\mid m$ ) under a large-sieve hypothesis, based on the first-order approximation $\Phi\approx 1/L$ (see (2.18)). Now, in §10.2, we will use Hölder in a similar spirit to prove Theorem 1.3. This relies crucially on several new features, including the more precise $\Phi$ -data captured in $\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})$ (compared to $1/L$ in [wang2023_large_sieve_diagonal_cubic_forms]).

For the next four results, let $C\in\mathcal{D}$ , let $\bm{N}\in\mathscr{R}_{17}(X)$ , let $\bm{t}\in\mathbb{R}^{2}$ , and let $\mathcal{S}\subseteq\mathcal{S}_{1}\cap[-C,C]^{m}$ . Roughly speaking, (10.11) will be useful when $C\leq X^{1/2-\delta}$ ; and otherwise, (10.9) will be useful for small $N_{0}$ (relative to $XC/N$ ), and (10.13) useful for larger $N_{0}$ .

Lemma 10.1.

Assume Conjecture 7.4. Then for some $A_{14}=A_{14}(F,\epsilon)>0$ , we have

(10.9)

\lVert\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\cdot\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\rVert_{\ell^{1}_{\bm{c}}(\mathcal{S})}\ll_{F,\epsilon}(1+\lvert t_{1}\rvert)^{A_{14}}\cdot\lvert\mathcal{S}\rvert^{1/2-\epsilon}(C^{m}+N_{1}^{m/3})^{1/2+\epsilon}N_{0}^{m/2+\epsilon}N^{1/2}.

Proof.

Plugging the trivial bound $\lvert S^{\natural}_{\bm{c}}(n_{0})\rvert\leq n_{0}^{(1+m)/2}$ into (10.4) gives

(10.10)

\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\ll_{\nu_{2}}N_{0}^{(1+m)/2}\cdot\lvert\mathcal{N}_{\bm{c}}\cap[N_{0},2N_{0})\rvert

for all $\bm{c}\in\mathcal{S}$ . So by Lemma 6.17 (with $A=(2-\epsilon)/\epsilon$ ), Conjecture 7.4 (with $C+N_{1}^{1/3}$ , $N_{1}$ in place of $Z$ , $N$ ), and Hölder over $\bm{c}\in\mathcal{S}$ , the left-hand of (10.9) is

\ll_{F,\epsilon}N_{0}^{(1+m)/2}\cdot(C^{m}N_{0}^{\epsilon})^{\epsilon/(2-\epsilon)}\cdot\lvert\mathcal{S}\rvert^{(1-2\epsilon)/(2-\epsilon)}\cdot[(1+\lvert t_{1}\rvert)^{A_{6}(F,\epsilon)}(C^{m}+N_{1}^{m/3})N_{1}^{(2-\epsilon)/2}]^{1/(2-\epsilon)}

for all $\epsilon\in(0,\frac{1}{2})$ , since $1=\frac{\epsilon}{2-\epsilon}+\frac{1-2\epsilon}{2-\epsilon}+\frac{1}{2-\epsilon}$ . Writing $N_{0}^{(1+m)/2}N_{1}^{1/2}=N_{0}^{m/2}N^{1/2}$ gives (10.9), since $\frac{1-2\epsilon}{2-\epsilon}\geq\frac{1}{2}-O(\epsilon)$ and $\lvert\mathcal{S}\rvert\leq C^{m}\leq C^{m}+N_{1}^{m/3}$ . ∎

Lemma 10.2.

Assume Conjectures 1.2 and 9.6. Then

(10.11)

\lVert\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\cdot\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\rVert_{\ell^{1}_{\bm{c}}(\mathcal{S})}\ll_{F,\epsilon}(1+\lvert t_{1}\rvert)^{\epsilon}\cdot\lvert\mathcal{S}\rvert^{1/2}(C^{m}+N_{0}^{m/3})^{1/2+\epsilon}N^{1/2+\epsilon}.

Proof.

By Proposition 3.2(8) and partial summation, $\lvert\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\rvert\ll_{\nu_{2},\epsilon}C^{\epsilon}(1+\lvert t_{1}\rvert)^{\epsilon}N_{1}^{1/2+\epsilon}$ . By Cauchy–Schwarz and (9.11), the left-hand side of (10.11) is $\ll_{F,\epsilon}[N_{0}\cdot\Sigma_{14}^{1,2}(C,N_{0})]^{1/2}\cdot\lvert\mathcal{S}\rvert^{1/2}\cdot C^{\epsilon}(1+\lvert t_{1}\rvert)^{\epsilon}N_{1}^{1/2+\epsilon}$ . Now use Conjecture 9.6 (with $C+N_{0}^{1/3}$ , $N_{0}$ in place of $Z$ , $N$ ). ∎

Proposition 10.3.

Assume Conjecture 9.8 with some $W$ . Then for any real $\delta\geq 0$ , we have

(10.12)

\sum_{\bm{c}\in\mathcal{S}:\,W(\bm{c})\neq 0}\lvert N_{0}^{-1/2}\cdot\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\rvert^{2+\delta}\ll_{F,\delta}(C^{m}+N_{0}^{m/3})N_{0}^{m\delta/2-9\eta_{10}(F,2)/10}.

Proof.

We may assume $\mathcal{S}\subseteq\{W\neq 0\}$ . By Hölder over $\mathcal{S}$ , the left-hand side of (10.12) is

\leq\lVert(N_{0}^{-1/2}\cdot\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t}))^{\delta+\epsilon}\rVert_{\ell^{2/\epsilon}_{\bm{c}}(\mathcal{S})}\cdot\lVert(N_{0}^{-1/2}\cdot\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t}))^{2-\epsilon}\rVert_{\ell^{2/(2-\epsilon)}_{\bm{c}}(\mathcal{S})},

for any $\epsilon\in[0,\delta]$ , since $\frac{\epsilon}{2}+\frac{2-\epsilon}{2}=1$ . The first factor here is $\ll_{\delta,\epsilon}(N_{0}^{m/2})^{\delta+\epsilon}(C^{m}N_{0}^{\epsilon})^{\epsilon/2}$ by (10.10) and Lemma 6.17; the second factor is $\ll_{\epsilon}[(C+N_{0}^{1/3})^{m}N_{0}^{-\eta_{10}(F,2)}]^{(2-\epsilon)/2}$ by Conjecture 9.8. Taking $\epsilon$ sufficiently small in terms of $m$ , $\eta_{10}(F,2)$ gives (10.12). ∎

Lemma 10.4.

Fix $\xi\in\mathbb{R}_{\geq 0}$ . Assume Conjectures 1.2, 9.6, and 9.8. Also assume Conjecture 7.4 if $\xi=0$ . Assume $N\ll C^{6}$ . Then for some $A_{15}=A_{15}(F,\epsilon)>0$ , we have

(10.13)

\lVert\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\cdot\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\rVert_{\ell^{1}_{\bm{c}}(\mathcal{S})}\ll_{F,\epsilon}(1+\lvert t_{1}\rvert)^{A_{15}}\cdot\frac{(CN_{1})^{\xi}(C^{m})^{1/2-\epsilon}(C^{m}+N^{m/3})^{1/2+\epsilon}N^{1/2}}{\min(C^{9/20},N_{0}^{4\eta_{10}(F,2)/5})}.

Proof.

Case 1: $\mathcal{S}\subseteq\{W=0\}$ . Then $\lvert\mathcal{S}\rvert\ll_{W}C^{m-1}$ (see e.g. [bhargava2014geometric]*Lemma 3.1). Inserting this into (10.11), we get (10.13) upon writing $C^{(m-1)/2}N^{\epsilon}\ll_{\epsilon}C^{(m-1)/2+6\epsilon}$ (for a small $\epsilon$ ).

Case 2: $\mathcal{S}\subseteq\{W\neq 0\}$ . Since $1=\frac{1-\epsilon}{2-\epsilon}+\frac{1}{2-\epsilon}$ , we may use (10.12) (with $\delta=\frac{\epsilon}{1-\epsilon}$ ), Conjecture 7.4 (if $\xi=0$ ) or GRH (if $\xi>0$ , using (7.17) and Proposition 3.2(8) to prove Conjecture 7.4 up to a factor of $Z^{\epsilon}$ ), and Hölder over $\bm{c}\in\mathcal{S}$ to bound the left-hand side of (10.13) by $O_{F,\epsilon}(N^{1/2}(CN_{1})^{\xi})\cdot[(C^{m}+N_{0}^{m/3})N_{0}^{m\delta/2-9\eta_{10}(F,2)/10}]^{(1-\epsilon)/(2-\epsilon)}\cdot[(1+\lvert t_{1}\rvert)^{A_{6}(F,\epsilon)}(C^{m}+N_{1}^{m/3})]^{1/(2-\epsilon)}$ . Since $C^{m}+N_{j}^{m/3}\ll C^{m}$ for some $j\in\{0,1\}$ , we get (10.13) if $\epsilon$ is small.

Case 3: The general case. Decompose $\mathcal{S}$ as $(\mathcal{S}\cap\{W=0\})\cup(\mathcal{S}\setminus\{W=0\})$ . ∎

Theorem 10.5.

Fix $\xi\in\mathbb{R}_{\geq 0}$ . Assume (2.6) and $2\mid m$ . Assume Conjectures 1.2, 9.6, and 9.8. Also assume Conjecture 1.4 if $\xi=0$ . Then $\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{F,w}X^{(6-m)/4+\xi}$ ; in fact,

(10.14)

\sum_{\bm{c}\in\mathcal{S}_{1}}\,\Bigl{\lvert}{\sum_{n\geq 1}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)J_{\bm{c},X}(n)}\Bigr{\rvert}\ll_{F,w}X^{(6-m)/4+\xi}.

Moreover, for any $X\in\mathbb{R}_{\geq 1}$ and $P_{0},P\in\mathbb{R}_{>0}$ , we have (for some constant $\eta_{11}=\eta_{11}(F)>0$ )

(10.15)

\int_{\bm{N}\in\mathscr{R}_{17,P_{0},P}(X)}d^{\times}\bm{N}\int_{\bm{t}\in\mathbb{R}^{2}}d\bm{t}\sum_{\bm{c}\in\mathcal{S}_{1}}\lvert g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\rvert\prod_{0\leq j\leq 1}\lvert\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t})\rvert\ll_{F,w}\frac{X^{(6-m)/4+\xi}}{\min(P_{0}P,X)^{\eta_{11}}},

where $\mathscr{R}_{17,P_{0},P}(X)\colonequals\{\bm{N}\in\mathscr{R}_{17}(X):N_{0}\geq P_{0}/2,\;N\leq A_{0}Y/P\}$ .

Proof.

By (10.2) and (10.5), the bound (10.15) (with $P_{0}=1$ and $P=1$ ) implies (10.14). And by (10.1), the bound (10.14) implies $\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{F,w}X^{(6-m)/4+\xi}$ . It remains to prove (10.15); for this, we may assume $P_{0}\geq 1$ and $P\geq 1$ (since if $P_{0}<1$ or $P<1$ , we may increase $P_{0}$ or $P$ while keeping the set $\mathscr{R}_{17,P_{0},P}(X)$ the same). To begin, we decompose $\mathcal{S}_{1}$ using (10.7), and then plug in (10.6) (with $\bm{\alpha}=\bm{0}$ ); this bounds the left-hand side of (10.15) by $O_{F,w,b}(1)$ times

(10.16)

\int_{\mathscr{R}_{17,P_{0},P}(X)}d^{\times}\bm{N}\int_{\mathbb{R}^{2}}d\bm{t}\sum_{C\in\mathcal{D},\;\lambda\in\mathcal{D}_{2}(C)}f(\bm{N},\bm{t},C,\lambda),

where we write (for convenience) $\mathcal{D}_{2}(C)=\{\lambda\in\mathcal{D}_{2}:1/C^{\deg{\Delta}}\leq\lambda\leq A_{13}\}$ and

(10.17)

f(\bm{N},\bm{t},C,\lambda)=\frac{\lVert\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})\cdot\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})\rVert_{\ell^{1}_{\bm{c}}(\mathcal{S}_{1}(C,\lambda))}\cdot N^{-1/2}(N+XC)^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}(1+C/X^{1/2})^{b}(1+XC\lambda/N)^{b}}.

Next, note that if $\xi=0$ , then Conjecture 7.4 holds by Propositions 6.10 and 7.5. So (10.9), (10.11), (10.13) are all at our disposal, no matter what $\xi$ is. Let $\theta=\frac{1}{10m}$ and $\epsilon=\frac{\theta}{100m}$ , say. Suppose $\bm{N}\in\mathscr{R}_{17}(X)$ and $b\geq 3+\max(A_{14}(F,\epsilon),\epsilon,A_{15}(F,\epsilon))$ . Let $C\in\mathcal{D}$ and $\lambda\in\mathcal{D}_{2}(C)$ .

Plugging (10.8) into (10.9) (for $\mathcal{S}=\mathcal{S}_{1}(C,\lambda)$ ) and integrating (10.17) over $\bm{t}$ gives

\int_{\mathbb{R}^{2}}d\bm{t}\,f(\bm{N},\bm{t},C,\lambda)\ll\frac{[C^{m}(\lambda+C^{-1})^{1/\deg{\Delta}}]^{1/2-\epsilon}(C^{m}+N_{1}^{m/3})^{1/2+\epsilon}N_{0}^{m/2+\epsilon}}{(N+XC)^{m/2-1}(1+C/X^{1/2})^{b}(1+XC\lambda/N)^{b}}.

Let $\alpha=(1/2-\epsilon)/\deg{\Delta}$ . Summing over $\lambda\in\mathcal{D}_{2}(C)$ (using Lemma 5.3 with $r=\lambda$ , $\tau=\frac{XC}{N}$ , $a=0$ , and $q\in\{\alpha,0\}$ , noting that $0\leq\alpha<b$ and $(\lambda+C^{-1})^{\alpha}\ll\lambda^{\alpha}+C^{-\alpha}$ ), we get

\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\frac{C^{(1/2-\epsilon)m}[\min(\frac{N}{XC},1)^{\alpha}+C^{-(1-\epsilon)\alpha}](C^{m}+N_{1}^{m/3})^{1/2+\epsilon}N_{0}^{m/2+\epsilon}}{(N+XC)^{m/2-1}(1+C/X^{1/2})^{b}}.

Writing $\min(\frac{N}{XC},1)\leq\frac{N}{XC}$ and $(N+XC)^{m/2-1}\geq(XC)^{m/2-1}$ , and then summing over $C\in\mathcal{D}$ using Lemma 5.3 (with $r=C$ and $\tau=X^{-1/2}$ , noting that $(\frac{1}{2}-\epsilon)m>\alpha+(\frac{m}{2}-1)$ ), we get

\sum_{C\in\mathcal{D},\;\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\frac{[(\frac{N}{Y})^{\alpha}+X^{-(1-\epsilon)\alpha/2}](X^{m/2}+X^{(1/2-\epsilon)m/2}N_{1}^{(1/2+\epsilon)m/3})N_{0}^{m/2+\epsilon}}{Y^{m/2-1}},

since $Y=X\cdot X^{1/2}$ . But $N_{1}\ll X^{3/2}$ (by (10.5)), so the right-hand side is $\ll[(\tfrac{N}{Y})^{\alpha}+X^{-(1-\epsilon)\alpha/2}]\cdot N_{0}^{m/2+\epsilon}\cdot X^{(6-m)/4}$ . Letting $\mathscr{R}_{18}\colonequals\{\bm{N}\in\mathscr{R}_{17,P_{0},P}(X):N_{0}^{m+\epsilon}\leq\min(\tfrac{Y}{N},X^{(1-\epsilon)/2})^{\alpha/2}\}$ , and writing $N_{0}^{m/2+\epsilon}\leq N_{0}^{-m/2}\cdot\min(\tfrac{Y}{N},X^{(1-\epsilon)/2})^{\alpha/2}$ for each $\bm{N}\in\mathscr{R}_{18}$ , we get

\int_{\mathscr{R}_{18}}d^{\times}\bm{N}\sum_{C\in\mathcal{D},\;\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\int_{1/4}^{A_{0}Y/P}d^{\times}{N}\,[(\tfrac{N}{Y})^{\alpha/2}+X^{-(1-\epsilon)\alpha/4}]P_{0}^{-m/2}X^{(6-m)/4},

by integrating over $N_{0}\geq P_{0}/2$ for each fixed valued of $N_{0}N_{1}=N$ . Thus

(10.18)

\int_{\mathscr{R}_{18}}d^{\times}\bm{N}\sum_{C\in\mathcal{D},\;\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll[P^{-\alpha/2}+X^{-(1-2\epsilon)\alpha/4}]P_{0}^{-m/2}X^{(6-m)/4}.

On the other hand, discarding the factor $(1+XC\lambda/N)^{b}$ in (10.17) (using $1+XC\lambda/N\geq 1$ ), plugging $\lvert\mathcal{S}\rvert\ll C^{m}$ into (10.11) (for $\mathcal{S}=\bigcup_{\lambda\in\mathcal{D}_{2}(C)}\mathcal{S}_{1}(C,\lambda)$ ), and integrating over $\bm{t}$ gives

\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f(\bm{N},\bm{t},C,\lambda)\ll\frac{C^{m/2}(C^{m}+N_{0}^{m/3})^{1/2+\epsilon}N^{\epsilon}}{(N+XC)^{m/2-1}(1+C/X^{1/2})^{b}}.

Writing $(N+XC)^{m/2-1}\geq(XC)^{m/2-1}$ , and then summing over $1\leq C\leq X^{1/2-\theta}$ , we get

\sum_{C\in\mathcal{D}:\,1\leq C\leq X^{1/2-\theta}}\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\frac{(X^{1/2-\theta})^{m/2}((X^{1/2-\theta})^{m}+N_{0}^{m/3})^{1/2+\epsilon}N^{\epsilon}}{(X^{3/2-\theta})^{m/2-1}}.

But $N_{0}^{m/3}\ll X^{m/2}$ by (10.5); integrating the previous display over $\mathscr{R}_{17}(X)$ thus gives

(10.19)

\int_{\mathscr{R}_{17}(X)}d^{\times}\bm{N}\sum_{C\in\mathcal{D}:\,1\leq C\leq X^{1/2-\theta}}\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\frac{X^{(6-m)/4}X^{(m/2+2)\epsilon}}{X^{\theta}}\leq\frac{X^{(6-m)/4}}{X^{9\theta/10}}.

Now suppose $C>X^{1/2-\theta}$ ; then $C>X^{1/2-1/10}=X^{2/5}\gg N^{1/6}$ . Discarding $(1+XC\lambda/N)^{b}$ in (10.17), taking $\mathcal{S}=\bigcup_{\lambda\in\mathcal{D}_{2}(C)}\mathcal{S}_{1}(C,\lambda)$ in (10.13), and integrating over $\bm{t}$ gives

\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f(\bm{N},\bm{t},C,\lambda)\ll\frac{(CN_{1})^{\xi}\cdot C^{(1/2-\epsilon)m}(C^{m}+N^{m/3})^{1/2+\epsilon}\cdot(N+XC)^{1-m/2}}{\min(C^{9/20},N_{0}^{4\eta_{10}(F,2)/5})(1+C/X^{1/2})^{b}}.

Writing $(N+XC)^{1-m/2}\leq(XC)^{1-m/2}$ , and summing over $C$ using Lemma 5.3, gives

\sum_{C\in\mathcal{D}:\,C>X^{1/2-\theta}}\sum_{\lambda\in\mathcal{D}_{2}(C)}\int_{\mathbb{R}^{2}}d\bm{t}\,f\ll\frac{(X^{1/2}N_{1})^{\xi}\cdot X^{(1/2-\epsilon)m/2}(X^{m/2}+N^{m/3})^{1/2+\epsilon}\cdot Y^{1-m/2}}{\min(X^{9/50},N_{0}^{4\eta_{10}(F,2)/5})}.

Here $N_{1},N\ll X^{3/2}$ , so the right-hand side is $\ll X^{2\xi}\cdot X^{(6-m)/4}\cdot(X^{-9/50}+N_{0}^{-4\eta_{10}(F,2)/5})$ . Let $\mathscr{R}_{19}\colonequals\mathscr{R}_{17,P_{0},P}(X)\setminus\mathscr{R}_{18}$ . Each $\bm{N}\in\mathscr{R}_{19}$ satisfies $N_{0}^{m+\epsilon}>\min(\tfrac{Y}{N},X^{(1-\epsilon)/2})^{\alpha/2}$ , and thus $N_{0}^{-\eta_{10}(F,2)}<\min(\tfrac{Y}{N},X^{(1-\epsilon)/2})^{-\beta}\leq(\tfrac{N}{Y})^{\beta}+X^{-(1-\epsilon)\beta/2}$ , where $\beta=\frac{\eta_{10}(F,2)\alpha}{2(m+\epsilon)}$ . Thus the integral $\int_{\mathscr{R}_{19}}d^{\times}\bm{N}$ of (the left-hand side of) the previous display is

\ll\int_{\mathscr{R}_{17,P_{0},P}(X)}d^{\times}\bm{N}\,X^{2\xi}\cdot X^{(6-m)/4}\cdot(X^{-9/50}+N_{0}^{-2\eta_{10}(F,2)/5}[(\tfrac{N}{Y})^{2\beta/5}+X^{-(1-\epsilon)\beta/5}]),

which is $\ll(X^{\epsilon-9/50}+P_{0}^{-2\eta_{10}(F,2)/5}[P^{-2\beta/5}+X^{-(1-2\epsilon)\beta/5}])\cdot X^{(6-m)/4+2\xi}$ (by integrating first over $N_{0}\geq P_{0}/2$ when $N$ is fixed, and then integrating over $N\leq A_{0}Y/P$ ). This, when combined with (10.18) and (10.19), establishes (10.15) with $\eta_{11}=\min(\frac{\alpha}{2},\frac{0.9\alpha}{4},\frac{9\theta}{10},\frac{17}{100},\frac{2\eta_{10}(F,2)}{5},\frac{2\beta}{5},\frac{0.9\beta}{5})$ (where $\theta=\frac{1}{10m}$ , $\alpha\geq\frac{0.4}{\deg{\Delta}}$ , and $\beta\geq\frac{\eta_{10}(F,2)}{10m}$ ), after replacing $\xi$ with $\xi/2$ if $\xi>0$ . ∎

Remark 10.6.

Our use of Hölder above is fairly uniform (each of (10.9), (10.11), (10.13) being based on “approximately Cauchy–Schwarz”), but the input required could maybe be slightly relaxed by applying Hölder with more varied exponents. For instance, if $C\approx X^{1/2}$ and $N_{0}\geq X^{\delta}$ , one could work with $\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})$ in $\ell^{1}$ and $\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})$ in $\ell^{\infty}$ (the $N_{0}^{\delta}$ saving from the former drowning out the $X^{\epsilon}$ loss from the latter); cf. (10.13). And if $N_{0}\leq X^{\delta}$ , one might hope to work with $\Sigma_{17,\bm{N}}^{\bm{c},1}(\bm{t})$ in $\ell^{1}$ over $\bm{c}$ in a residue class to modulus $n_{0}\asymp N_{0}$ (with some nontrivial archimedean restrictions on $\bm{c}$ ), and then work with $\Sigma_{17,\bm{N}}^{\bm{c},0}(\bm{t})$ in $\ell^{1}$ afterwards.

Proof of Theorem 1.3.

Let $\mathcal{D}=\{1,2,4,8,\ldots\}$ . Let $T_{0,X}(\theta)\colonequals\sum_{\lvert x\rvert\leq X}e(\theta x^{3})$ for $\theta\in\mathbb{R}$ ; then

(10.20)

N_{F}(X)=\int_{[0,1]}d\theta\,\lvert T_{0,X}(\theta)\rvert^{6}=\lVert T_{0,X}(\theta)^{6}\rVert_{L^{1}_{\theta}([0,1])}.

Now let $T_{1,X}(\theta)\colonequals\sum_{\lvert x\rvert\in(X/2,X]}e(\theta x^{3})$ . Let $\lVert f(\lambda)\rVert_{\ell^{q}_{\lambda}(\mathcal{D})}\colonequals(\sum_{\lambda\in\mathcal{D}}\lvert f(\lambda)\rvert^{q})^{1/q}$ for $q\geq 1$ ; then

(10.21)

T_{0,X}(\theta)=1+\sum_{\lambda\in\mathcal{D}}T_{1,X/\lambda}(\theta)\ll 1+\lVert(X/\lambda)^{\rho}\rVert_{\ell^{6/5}_{\lambda}(\mathcal{D})}\cdot\lVert(X/\lambda)^{-\rho}T_{1,X/\lambda}(\theta)\rVert_{\ell^{6}_{\lambda}(\mathcal{D})}

(by Hölder), where $\rho=1/100$ , say. Since $\lVert(X/\lambda)^{\rho}\rVert_{\ell^{6/5}_{\lambda}(\mathcal{D})}\ll X^{\rho}$ , we deduce (upon inserting (10.21) into (10.20)) that if $\lVert T_{1,X/\lambda}(\theta)^{6}\rVert_{L^{1}_{\theta}([0,1])}\ll(X/\lambda)^{3}$ holds, then (since $6\rho<3$ )

(10.22)

N_{F}(X)\ll 1+X^{6\rho}\cdot\lVert(X/\lambda)^{-6\rho}T_{1,X/\lambda}(\theta)^{6}\rVert_{L^{1}_{(\lambda,\theta)}(\mathcal{D}\times[0,1])}\ll X^{3}.

Now let $w_{\star}(\bm{x})\colonequals\prod_{1\leq j\leq 6}\bm{1}_{\lvert x_{j}\rvert\in(1/2,1]}$ , and choose $w\in C^{\infty}_{c}((\mathbb{R}\setminus\{0\})^{6})$ with $w\geq w_{\star}$ . Then

(10.23)

\lVert T_{1,X}(\theta)^{6}\rVert_{L^{1}_{\theta}([0,1])}=N_{F,w_{\star}}(X)\leq N_{F,w}(X).

But $w$ satisfies (1.11) (and thus (2.6)). And we are assuming (for Theorem 1.3) Conjectures 1.2, 1.4, and 1.5; in particular, Conjectures 9.6 and 9.8 hold by Propositions 9.7 and 9.9, respectively. So Theorem 10.5 (with $\xi=0$ ) gives $\Sigma^{\natural}(X,\mathcal{S}_{1})\ll 1$ . But $\Sigma^{\natural}(X,\mathcal{S}_{0})\ll 1$ by Theorem 2.5. So by (2.10) (and the definitions (2.15), (1.6)), we have

N_{F,w}(X)/X^{3}=O_{A}(Y^{-A})+\Sigma^{\natural}(X,\mathcal{S}_{0})+\Sigma^{\natural}(X,\mathcal{S}_{1})\ll 1.

Hence $N_{F}(X)\ll X^{3}$ by (10.23), (10.22). Given (1.8), a standard Cauchy–Schwarz argument then leads to the desired application to $F_{0}(S^{3})$ (producing a “positive lower density”). ∎

10.3. Delta cancellation

To prove Theorems 1.6 and 1.9, we will complement (10.15) using Propositions 7.15 and 7.16, identifying $\bm{c}$ -cancellation in some pieces of $\Sigma^{\natural}(X,\mathcal{S}_{1})$ . Since $J_{\bm{c},X}(n)$ is not compactly supported in $\bm{c}$ , we begin with a decomposition resembling (8.18).

Recall $\nu_{0}$ , $\nu_{1}$ from §1.2. Let $\overline{\nu}_{0}\colonequals 1-\nu_{0}$ . For all $\kappa\in\mathbb{R}_{>0}$ , let (cf. (8.16), (8.17))

(10.24)		$\displaystyle\Sigma_{18,0}(X,\bm{N},\bm{t})$	$\displaystyle\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}}\nu_{0}(\bm{c}/X^{1/2})g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t}),$
(10.25)		$\displaystyle\Sigma_{18,1,\kappa}(X,\bm{N},\bm{t})$	$\displaystyle\colonequals\sum_{\bm{c}\in\mathcal{S}_{1}}\overline{\nu}_{0}(\bm{c}/X^{1/2})\nu_{1}(\bm{c}/\kappa)g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t}).$

We have $\Sigma_{18,1,\kappa}(X,\bm{N},\bm{t})=0$ unless there exists $\bm{c}\in\mathcal{S}_{1}$ satisfying $X^{1/2}/2<\lVert\bm{c}\rVert\leq m\kappa$ . Using $\int_{\kappa>0}d^{\times}{\kappa}\,\nu_{1}(\bm{c}/\kappa)=1$ (valid for $\bm{c}\in\mathbb{R}^{m}\setminus\{\bm{0}\}$ ), we may thus write (cf. (8.18))

(10.26)

\sum_{\bm{c}\in\mathcal{S}_{1}}g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t})=\Sigma_{18,0}(X,\bm{N},\bm{t})+\int_{X^{1/2}/2m}^{\infty}d^{\times}{\kappa}\,\Sigma_{18,1,\kappa}(X,\bm{N},\bm{t}).

Recall $\mathscr{R}_{17}(X)$ from (10.5). Let $\mathscr{R}_{17}^{P_{0},P}(X)\colonequals\{\bm{N}\in\mathscr{R}_{17}(X):N_{0}<P_{0}/2,\;N>A_{0}Y/P\}$ for reals $P_{0},P>0$ . In terms of $\mathscr{R}_{17,P_{0},P}(X)$ from Theorem 10.5, we have

(10.27)

\mathscr{R}_{17}^{P_{0},P}(X)\subseteq\mathscr{R}_{17}(X)\subseteq\mathscr{R}_{17}^{P_{0},P}(X)\cup\mathscr{R}_{17,1,P}(X)\cup\mathscr{R}_{17,P_{0},1}(X).

By (10.1), (10.2), (10.5), and (10.27), the bound (10.15) (when applicable) implies

(10.28)

\Sigma^{\natural}(X,\mathcal{S}_{1})-\int_{\mathscr{R}_{17}^{P_{0},P}(X)}d^{\times}\bm{N}\int_{\mathbb{R}^{2}}d\bm{t}\sum_{\bm{c}\in\mathcal{S}_{1}}g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t})\ll\frac{X^{(6-m)/4+\xi}}{\min(P_{0},P,X)^{\eta_{11}}}.

This leads to the following results.

Theorem 10.7.

Assume (2.6) and $2\mid m$ . Assume Conjectures 1.2, 1.4, 1.8, 9.6, and 9.8. Then for $X\geq 1$ , we have $\Sigma^{\natural}(X,\mathcal{S}_{1})=o_{F,w;X\to\infty}(X^{(6-m)/4})$ .

Proof.

Before proceeding, note that Conjecture 7.14 holds by Propositions 6.10, 6.13, and 7.15, since we assume Conjectures 1.2, 1.4, and 1.8. Let $M$ be a real number to be chosen later.

Let $\bm{N}\in\mathscr{R}_{17}^{P_{0},P}(X)$ . For each real $Z\geq 2X^{1/2}$ , let (in the context of (7.25))

\nu_{18,0}=\nu_{18,0,X,\bm{N},\bm{t},Z}=\nu_{0}(Z\bm{c}/X^{1/2})g_{Z\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\nu_{2}(r)

(noting that $\operatorname{Supp}{\nu_{0}}\subseteq[-2,2]^{m}$ , so $\nu_{18,0}$ is supported on $[-1,1]^{m}\times[1,2]$ ). Then after plugging (10.4) into (10.24) and decomposing $\mathcal{S}_{1}$ into residue classes modulo $n_{0}$ , we get

\Sigma_{18,0}(X,\bm{N},\bm{t})=\sum_{n_{0}\geq 1}\nu_{2}(n_{0}/N_{0})n_{0}^{-it_{0}}\sum_{1\leq\bm{a}\leq n_{0}:\,n_{0}\in\mathcal{N}_{\bm{a}}}S^{\natural}_{\bm{a}}(n_{0})\cdot\Sigma_{11}^{\bm{a},n_{0}}(\nu_{18,0},Z,N_{1})

(in terms of $\Sigma_{11}^{\bm{a},n_{0}}$ from (7.25)). By Conjecture 7.14 (with $\nu=\nu_{18,0}$ , and with $2X^{1/2}+N_{1}^{1/3}$ , $N_{1}$ in place of $Z$ , $N$ ) and the trivial bound $\lvert S^{\natural}_{\bm{a}}(n_{0})\rvert\leq n_{0}^{(1+m)/2}$ , we get

\Sigma_{18,0}(X,\bm{N},\bm{t})\ll N_{0}^{(3+m)/2}X^{m/2}N_{1}^{1/2}(o_{M\to\infty}(1)+o_{M;X\to\infty}(1))\cdot\mathcal{M}_{1,A_{7}}(\nu_{18,0}),

provided $\min(2X^{1/2},N_{1})\geq 2M\geq 2(2N_{0})^{10/9}$ . Additionally, by (10.6) (with $Z\bm{c}=(2X^{1/2}+N_{1}^{1/3})\bm{c}$ in place of $\bm{c}$ ) and the definition (6.24) of $\mathcal{M}_{1,k}$ (with $\nu=\nu_{18,0}$ , $k=A_{7}$ ), we have

\mathcal{M}_{1,A_{7}}(N^{1/2}\nu_{18,0})\ll_{b}\frac{(X^{1/2}/X^{1/2}+X\cdot X^{1/2}/N)^{1}\cdot(N+0)^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}(1+0)^{b}(1+0)^{b}}\ll\frac{(Y/N)\cdot N^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}}.

Now, for each $\kappa\geq X^{1/2}/2m$ and $Z\geq m\kappa$ (noting that $\operatorname{Supp}{\nu_{1}}\subseteq[-m,m]^{m}$ ), let

\nu_{18,1,\kappa}=\nu_{18,1,\kappa,X,\bm{N},\bm{t},Z}=\overline{\nu}_{0}(Z\bm{c}/X^{1/2})\nu_{1}(Z\bm{c}/\kappa)g_{Z\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\nu_{2}(r).

Applying Conjecture 7.14 with $m\kappa+N_{1}^{1/3}$ , $N_{1}$ in place of $Z$ , $N$ , we get

\Sigma_{18,1,\kappa}(X,\bm{N},\bm{t})\ll N_{0}^{(3+m)/2}\kappa^{m}N_{1}^{1/2}(o_{M\to\infty}(1)+o_{M;X\to\infty}(1))\cdot\mathcal{M}_{1,A_{7}}(\nu_{18,1,\kappa}),

provided $\min(X^{1/2}/2,N_{1})\geq 2M\geq(2N_{0})^{10/9}$ . This time, since $\bm{0}\notin\operatorname{Supp}{\nu_{1}}$ , the bound (10.6) (with $(m\kappa+N_{1}^{1/3})\bm{c}$ in place of $\bm{c}$ ) and (6.24) (with $\nu=\nu_{18,1,\kappa}$ , $k=A_{7}$ ) give

\mathcal{M}_{1,A_{7}}(N^{1/2}\nu_{18,1,\kappa})\ll_{b}\frac{(\kappa/X^{1/2}+X\kappa/N)^{1}\cdot(N+X\kappa)^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}(1+\kappa/X^{1/2})^{b}(1+0)^{b}}\ll\frac{(Y/N)\cdot(X\kappa)^{1-m/2}}{(1+\lVert\bm{t}\rVert)^{b}(\kappa/X^{1/2})^{b-1}}.

Inserting our bounds on $\Sigma_{18,0}$ , $\Sigma_{18,1,\kappa}$ into (10.26), assuming $b\geq 3+m/2$ , we find that if $\min(X^{1/2}/2,N_{1})\geq 2M\geq(2N_{0})^{10/9}$ , then the left-hand side of (10.26) is

(10.29)

\ll N_{0}^{(2+m)/2}X^{m/2}(o_{M\to\infty}(1)+o_{M;X\to\infty}(1))\cdot(Y/N)\cdot N^{1-m/2}/(1+\lVert\bm{t}\rVert)^{b}.

Since $N_{0}<P_{0}/2$ and $N_{1}=N/N_{0}>A_{0}Y/P_{0}P$ , we conclude (upon integrating over $\bm{t}\in\mathbb{R}^{2}$ , then over $N_{0}<P_{0}/2$ with $N_{0}N_{1}=N$ fixed, and finally over $A_{0}Y/P<N\leq A_{0}Y$ ) that

\int_{\mathscr{R}_{17}^{P_{0},P}(X)}d^{\times}\bm{N}\int_{\mathbb{R}^{2}}d\bm{t}\sum_{\bm{c}\in\mathcal{S}_{1}}g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t})\ll\frac{P_{0}^{(2+m)/2}X^{m/2}o_{M\to\infty}(1)\cdot Y}{(Y/P)^{m/2}},

provided $X\gg_{M,P_{0},P}1$ and $M\gg_{P_{0}}1$ hold with large enough implied constants. Therefore, there exist functions $f_{3},f_{4}\colon\mathbb{R}_{>0}\to\mathbb{Z}_{\geq 1}$ such that if $M\geq f_{3}(P_{0}+P)$ and $X\geq f_{4}(M)$ , then the left-hand side of the previous display is $o_{P_{0}+P\to\infty}(Y^{1-m/2}X^{m/2})$ . It follows from (10.28) (with $\xi=0$ ) that if $X\geq f_{4}(f_{3}(P_{0}+P))$ , then $\Sigma^{\natural}(X,\mathcal{S}_{1})=o_{\min(P_{0},P,X)\to\infty}(X^{(6-m)/4})$ . Finally, suppose $X\geq f_{4}(f_{3}(2))$ , let $P$ be the largest integer in $[1,X]$ with $f_{4}(f_{3}(2P))\leq X$ (so that $P\to\infty$ as $X\to\infty$ ), and take $P_{0}=P$ , to get $\Sigma^{\natural}(X,\mathcal{S}_{1})=o_{X\to\infty}(X^{(6-m)/4})$ . ∎

Proof of Theorem 1.6.

Unconditionally, by (1.3), (2.10), and (2.16), we have

(10.30)

E_{F,w}(X)/X^{3}=O(X^{2.75+\epsilon})/X^{3}+\Sigma^{\natural}(X,\mathcal{S}_{1}).

Now assume (1.11) (so (2.6) holds). Then Theorem 10.7 gives $\Sigma^{\natural}(X,\mathcal{S}_{1})=o_{X\to\infty}(X^{(6-m)/4})$ , since Conjectures 9.6 and 9.8 hold by Propositions 9.7 and 9.9, respectively (since we assume Conjecture 1.5). Upon plugging this bound into (10.30), we get (1.5).

The Hasse principle for $V$ follows from (1.5), upon choosing $w$ with $\sigma_{\infty,F,w}>0$ (possible since $V(\mathbb{R})$ contains a point with $x_{1}\cdots x_{6}\neq 0$ ). Now suppose $F=x_{1}^{3}+\dots+x_{6}^{3}$ . Then by [wang2023prime]*Theorem 1.1 (or [wang2022thesis]*Theorem 2.1.8), we find (from (1.5)) that $F_{0}(\mathbb{Z}^{3})$ has density $1$ in $\{a\in\mathbb{Z}:a\not\equiv\pm 4\bmod{9}\}$ . (See [wang2023prime] for details on this last deduction, which is based on [diaconu2019admissible]. Diaconu assumes an analog of (1.5) over rather quantitatively deformed regions $R^{\ast}$ , whereas we work with fixed weights $w$ . It would be very interesting to see if there could be any miraculous cancellation or symmetries in the analog of $J_{\bm{c},X}(n)$ over $R^{\ast}$ , but at the moment it seems easier to handle minimally deformed regions.) ∎

Proof of Corollary 1.7.

(Here we drop the assumption (2.6).) Let $\rho>0$ be a parameter tending to $0$ slowly as $X\to\infty$ . Use (1.8) and Hölder’s inequality to upper bound the contribution to $N_{F,w}(X)$ from points $\bm{x}\in\mathbb{Z}^{m}$ with $\min(\lvert x_{1}\rvert,\dots,\lvert x_{m}\rvert)\leq\rho\cdot X$ . Then use Theorem 1.6 to estimate the remaining contribution to $N_{F,w}(X)$ . This gives (1.5) for arbitrary $w\in C^{\infty}_{c}(\mathbb{R}^{m})$ . Hooley’s conjecture follows upon taking a suitable sequence of weights $w$ . ∎

Theorem 10.8.

Assume (2.6) and $2\mid m$ . Assume Conjectures 1.2, 1.10, 1.11, 9.6, and 9.8. Then for some constant $\eta_{12}=\eta_{12}(F)>0$ , we have $\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{F,w}X^{(6-m)/4-\eta_{12}}$ .

Proof.

Proposition 7.16 applies, since we assume Conjectures 1.2, 1.10, 1.11. We now mimic the proof of Theorem 10.7, while replacing each use of Conjecture 7.14 with Proposition 7.16.

Let $\bm{N}\in\mathscr{R}_{17}^{P_{0},P}(X)$ . Proposition 7.16 implies that for any real $M\in[1,X^{\eta_{2}/2}]$ satisfying $\min(2X^{1/2},N_{1})\geq 2M\geq 2(2N_{0})^{2}$ , we have

\Sigma_{18,0}(X,\bm{N},\bm{t})\ll_{\epsilon}N_{0}^{(3+m)/2}X^{m/2+\epsilon}N_{1}^{1/2}(M^{1/6\deg{H}}N_{1}^{-1/6}+M^{-\eta_{5}/4\deg{H}})\mathcal{M}_{1,A_{8}}(\nu_{18,0}).

Furthermore, for each $\kappa\geq X^{1/2}/2m$ , Proposition 7.16 implies that

\Sigma_{18,1,\kappa}(X,\bm{N},\bm{t})\ll_{\epsilon}N_{0}^{(3+m)/2}\kappa^{m+\epsilon}N_{1}^{1/2}(M^{1/6\deg{H}}N_{1}^{-1/6}+M^{-\eta_{5}/4\deg{H}})\mathcal{M}_{1,A_{8}}(\nu_{18,1,\kappa}).

for any real $M\in[1,(X^{1/2}/2)^{\eta_{2}}]$ satisfying $\min(X^{1/2}/2,N_{1})\geq 2M\geq 2(2N_{0})^{2}$ .

Now let $M=X^{\min(1,\eta_{2})/2}/(4+2^{\eta_{2}})$ , and suppose

(10.31)

(A_{0}Y/P_{0}P)^{1/2}\geq 2M\geq 2P_{0}^{2},

so that $N_{1}\gg M^{2}$ and (thus) $M^{1/6\deg{H}}N_{1}^{-1/6}+M^{-\eta_{5}/4\deg{H}}\ll M^{-\min(1,\eta_{5})/6\deg{H}}$ . Arguing as we did for (10.29), we find (assuming $b\geq 3+m/2$ ) that the left-hand side of (10.26) is

\ll_{\epsilon}N_{0}^{(2+m)/2}X^{m/2+\epsilon}(M^{-\min(1,\eta_{5})/6\deg{H}})\cdot(Y/N)\cdot N^{1-m/2}/(1+\lVert\bm{t}\rVert)^{b}.

So (upon integrating over $\bm{t}$ , then over $N_{0}$ , and finally over $N$ )

\int_{\mathscr{R}_{17}^{P_{0},P}(X)}d^{\times}\bm{N}\int_{\mathbb{R}^{2}}d\bm{t}\sum_{\bm{c}\in\mathcal{S}_{1}}g_{\bm{c},X,\bm{N}}^{\vee}(i\bm{t})\prod_{0\leq j\leq 1}\Sigma_{17,\bm{N}}^{\bm{c},j}(\bm{t})\ll_{\epsilon}\frac{P_{0}^{(2+m)/2}X^{m/2+\epsilon}\cdot Y}{M^{\min(1,\eta_{5})/6\deg{H}}\cdot(Y/P)^{m/2}}.

Now let $\gamma=\min(1,\eta_{5})/12\deg{H}\in(0,1/12]$ and $P_{0}=M^{\gamma/(2+m)}$ , $P=A_{0}M^{\gamma/m}/4$ . Then (10.31) holds (since $\gamma\leq 1/2$ and $Y\geq M^{3}$ ), and the right-hand side of the previous display is $\ll M^{\gamma/2+\gamma/2-2\gamma}X^{m/2+\epsilon}Y^{1-m/2}=M^{-\gamma}X^{(6-m)/4+\epsilon}$ . So by (10.28) (with $\xi=\epsilon>0$ ) we have

\Sigma^{\natural}(X,\mathcal{S}_{1})\ll_{\epsilon}X^{(6-m)/4+\epsilon}(M^{-\gamma}+P_{0}^{-\eta_{11}})\ll X^{(6-m)/4-\eta_{12}},

where $\eta_{12}=\frac{9}{10}\cdot\min(1,\eta_{11}/(2+m))\cdot\gamma\cdot\min(1,\eta_{2})/2$ , say. ∎

Proof of Theorem 1.9.

Proceed as in the proof of Theorem 10.7, but use Theorem 10.8 instead of Theorem 10.7, to get (from (10.30)) the bound $E_{F,w}(X)/X^{3}\ll_{\epsilon}X^{-0.25+\epsilon}+X^{-\eta_{12}}$ . ∎

Acknowledgements

Many thanks to Amit Ghosh and Peter Sarnak for suggesting the main problem addressed here. I also thank my advisor, Peter Sarnak, for his guidance and encouragement.^†^†This work was partially supported by NSF grant DMS-1802211. I thank Calvin Deng and Yotam Hendel for sharing references for (4.1) and (4.2), and Yotam for discussions related to §9. For early comments, I thank Manjul Bhargava, Andy Booker, Tim Browning, Brian Conrey, Simona Diaconu, Bill Duke, Roger Heath-Brown, Nick Katz, Will Sawin, Bob Vaughan, and Trevor Wooley. Thanks also to everyone listed in [wang2022thesis], and to people from seminars, conferences, and other events. For more recent interactions, I thank Louis-Pierre Arguin, Emma Bailey, Paul Bourgade, Alex Gamburd, Jayce Getz, Jeff Hoffstein, Junehyuk Jung, Victor Kolyvagin, Valeriya Kovaleva, Lillian Pierce, Kannan Soundararajan, Yuri Tschinkel, Katy Woo, Max Xu, Liyang Yang, and Peter Zenz. Finally, I thank my family for their exceptional support.

Sums of cubes and the Ratios Conjectures

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

Theorem (Hooley; Heath-Brown).

Theorem.

Theorem 1.1 ([hooley1986HasseWeil, hooley_greaves_harman_huxley_1997]; [heath1998circle]).

Conjecture 1.2 (HW2).

Theorem 1.3.

Conjecture 1.4 (R2’).

Conjecture 1.5 (SFSCp,3).

Theorem 1.6.

Corollary 1.7.

Conjecture 1.8 (RA1oo).

Theorem 1.9.

Conjecture 1.10 (RA1δ\delta).

Conjecture 1.11 (EKL).

1.1. Proof overview

1.2. Conventions

2. Background on discriminants and delta

Lemma 2.1.

Proof.

Theorem 2.2 ([buse2014discriminant]*Corollary 4.30).

Corollary 2.3.

Proof.

Proposition 2.4.

Proof.

Theorem 2.5 ([wang2023_isolating_special_solutions]).

Proof.

Conditional proof sketch for (2.17).

3. Background on individual LL-functions

3.1. Geometric background

Remark 3.1.

3.2. Automorphic background

Proposition 3.2.

Proof.

4. Local control on polynomials and LL-functions

Proposition 4.1.

Proof.

Corollary 4.2.

Proof.

Proposition 4.3 ([kisin1999local]).

Proof.

Lemma 4.4.

Proof.

Remark 4.5.

5. General separation technique

Proposition 5.1 (Standard Mellin bound).

Proof.

Lemma 5.2 (“Dyadic partial Mellin summation”).

Proof.

Lemma 5.3.

Proof.

6. Statistics of families of LL-functions

6.1. Computing local averages

Proposition 6.1 (LocAv).

Proof.

6.2. The Sarnak–Shin–Templier framework

Proposition 6.2.

Proof.

6.3. The Ratios Recipe

Conjecture 6.3 (R2oo).

Remark 6.4.

Remark 6.5.

Remark 6.6.

6.3.1. Deriving (RA1)

Remark 6.7.

6.3.2. Deriving (R2)

6.4. From (R2) to (R2’) and (R2’E)

Proposition 6.8.

Proof.

Conjecture 6.9 (R2’E).

Proposition 6.10.

Proof.

6.5. From (RA1) to (RA1’E)

Definition 6.11.

Conjecture 6.12 (RA1oo’E).

Proposition 6.13.

Proof.

Conjecture 1.5 (SFSC_p,3).

Conjecture 1.8 (RA1 $o$ ).

Conjecture 1.10 (RA1 $\delta$ ).

3. Background on individual $L$ -functions

4. Local control on polynomials and $L$ -functions

6. Statistics of families of $L$ -functions

Conjecture 6.3 (R2 $o$ ).

Conjecture 6.12 (RA1 $o$ ’E).

Proposition 6.14 (RA1 $\delta$ ’E).

Proposition 6.18 ( $\bigwedge$ 2E).

7. Adapting $L$ -function statistics to delta

Corollary 7.3 ( $\Phi$ 3E).

Conjecture 7.14 (RA1 $o$ ’E’).

Proposition 7.16 (RA1 $\delta$ ’E’).

Conjecture 9.4 (SFSC_q,3).