On a conjecture of Soundararajan

Let $P(n)$ denote the largest prime factor of the natural number $n\geqslant 2$, and put $P(1)\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=1$. A number $n$ is said to be $y$-smooth if $n$ has no prime factor exceeding $y$, that is, if $P(n)\leqslant y$.

Let $\mathcal{S}(y)$ denote the set of all $y$-smooth numbers, and let $\mathcal{S}(x,y)$ be the set of $y$-smooth numbers not exceeding $x$:

As usual, we use $\Psi(x,y)$ to denote the cardinality of $S(x,y)$.

In this note, we study the distribution of smooth numbers over arithmetic progressions $a\bmod q$ with the coprimality condition $(a,q)=1$. Defining

it is expected that the asymptotic relation

holds (with $\varphi$ the Euler function) under the condition $(a,q)=1$, over a wide range in the parameters $x,y,q,a$.

Soundararajan[22, Conjecture I(A)] has proposed the following conjecture.

Earlier, Granville [6, 7] had established this result for $A<1$, and he pointed out that the proof for arbitrarily large values of $A$ must lie fairly deep, for it implies Vinogradov’s conjecture that the least quadratic nonresidue modulo $p$ is of size $p^{o(1)}$. Soundararajan [22] has shown that (1.1) holds for moduli

provided that

In [10] Harper demonstrates how to remove the conditions (1.2), thereby settling Soundararajan’s Conjecture for all $A<4\sqrt{e}$.

Short of an improvement of the Burgess bound on character sums, one can consider the following variant of Soundararajan’s Conjecture in which the moduli all belong to a prescribed subset $\mathcal{Q}$ of the natural numbers $\mathbb{N}$.

Note that Soundararajan’s Conjecture is nothing but ${\tt SC}\llbracket\hskip 1.0pt\mathbb{N}\hskip 1.0pt\rrbracket$, and it is clear that ${\tt SC}\llbracket\hskip 1.0pt\mathbb{N}\hskip 1.0pt\rrbracket$ holds if and only if ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q}\hskip 1.0pt\rrbracket$ is true for every set $\mathcal{Q}\subseteq\mathbb{N}$. We say that Soundararajan’s Conjecture holds over $\mathcal{Q}$ whenever ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q}\hskip 1.0pt\rrbracket$ is true.

We start by establishing ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q}\hskip 1.0pt\rrbracket$ for a class of sufficiently smooth moduli.

An important special case of Corollary 1.2 is the set $\mathcal{Q}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=p^{\mathbb{N}}$ consisting of all powers of a fixed prime $p$. In fact, our work in the present paper has been initially motivated by a series of results on arithmetic problems involving progressions modulo large powers of a fixed prime. Such results include:

• •

  bounds on the zero-free regions of $L$-functions, which leads to results on the distribution of primes in arithmetic progressions (see [1, 2, 5, 14]),

• •

  asymptotic formulas in the Dirichlet problem on sums with the divisor function over arithmetic progressions modulo $p^{n}$ (see [16, 19]),

• •

  asymptotic formulas for moments of $L$-functions (see [20, 21]).

It is clear that, for any given set $\mathcal{Q}$, to establish ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q}\hskip 1.0pt\rrbracket$ one must show that the following hypothesis holds for all large $A$.

Since the number of moduli $q\leqslant x$ with $q\geqslant P(q)^{Q_{A}}$ is $\Psi(x,x^{1/Q_{A}})\sim\rho(Q_{A})x$, where $\rho$ is the Dickman function, Theorem 1.1 implies that the following variant of Soundararajan’s Conjecture (with $A$ arbitrary but fixed) holds over a set of positive asymptotic density.

Corollary 1.3 complements certain Bombieri-Vinogradov type results due to Granville and Shao [8] and Harper [11], which imply (1.1) for a set of moduli $q$ of asymptotic density one, but in more restrictive ranges of $y$. For example, none of those results apply to very smooth numbers with (say) $y$ of size $(\log x)^{o(1)}$.

The next result asserts that the asymptotic relation (1.1) holds as $q$ varies over a set of exceptional moduli.

In particular, Corollary 1.5 shows that any future work on Soundararajan’s Conjecture (over $\mathbb{N}$) can assume that Siegel zeros do not exist.

Our proofs of Theorems 1.1 and 1.4 rely on the argument of Harper [10] (which in turn builds upon original ideas of Soundararajan [22]). The treatment of the so-called “problem range” is the primary issue (see §2.2 below), thus a major part of the proof of Theorem 1.1 is devoted to elimination of this range. This is accomplished via a combination of results of Chang [3] and Iwaniec [14], which give bounds on certain character sums and on the zero-free regions of $L$-functions modulo highly composite integers. We remark that, for a slightly more restrictive class of moduli, some stronger bounds have been obtained by the authors (see [1, 2]), but these do not lead to better results on Soundararajan’s Conjecture. Concerning Theorem 1.4, our proof exploits the “Deuring-Heilbronn” phenomenon on the repelling nature of zeros of $L$-functions close to one.

In what follows, given functions $F$ and $G>0$ we use the equivalent notations $F=O(G)$ and $F\ll G$ to signify that the inequality $|F|\leqslant c\,G$ holds with some constant $c>0$. Throughout the paper, any implied constants may depend on the parameters $A$ and $\Phi$ but are independent of other variables.

We also write $F\asymp G$ or $F=\Theta(G)$ whenever $F,G>0$ and we have both $F=O(G)$ and $G=O(F)$.

The notations $F\sim G$ and $F=o(G)$ are used to indicate that $F/G\to 1$ and $F/G\to 0$, respectively, as certain specified parameters tend to infinity.

Harper’s theorem [10, Theorem 1] implies ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q},A\hskip 1.0pt\rrbracket$ for any set $\mathcal{Q}$ and any $A<4\sqrt{e}$. Thus, in what follows we can assume that $A\geqslant 4\sqrt{e}$ and $y^{4\sqrt{e}}\leqslant q\leqslant y^{A}$. In particular, the parameters

are comparable in size (that is, $u\asymp v$) since $A\geqslant u/v\geqslant 4\sqrt{e}$, and so $u\to\infty$ if and only if $v\to\infty$.

We remark that, in this section and the next, $y$ is sometimes required to exceed a large number that might depend on $A$. However, in view of (1.4), we can begin by taking $Q_{A}$ large enough to guarantee that $y$ meets these requirements.

For any character $\chi$ modulo $q$ we put

In particular, $\Psi_{q}(x,y)=\Psi(x,y;\chi_{0})$, where $\chi_{0}$ is the principal character. Using Dirichlet orthogonality we see that (1.1) is equivalent to the assertion that

As in [10, 22] it suffices to establish a smooth variant of (2.1). More precisely, let $\Phi:[0,\infty)\to[0,1]$ be a function that is supported on $[0,2]$, equal to one on $[0,\tfrac{1}{2}]$, and such that $\Phi\in C^{9}$, that is, $\Phi$ is nine times continuously differentiable. For every character $\chi$ modulo $q$ we denote

Then, it is enough show that

holds, since the passage from (2.2) back to (2.1) can be accomplished using the unsmoothing method outlined by Harper [10, Appendix A].

As in [10, 22] we start by writing

for any $c>0$, where

and

Note that the bound

follows from our smoothness assumption on $\Phi$ (using integration by parts and the continuity of $\Phi^{(9)}$). We choose $c$ to be $\alpha\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=\alpha(x,y)$, the unique positive solution to the equation

The quantity $\alpha$ is introduced in a saddle point argument of Hildebrand and Tenenbaum [12] for $\Psi(x,y)$ (see also [13]), and it has been applied by de la Bretéche and Tenenbaum [4] to $\Psi_{q}(x,y)$ for an arbitrary modulus $q$. Using only the trivial bound $\omega(q)\ll\log q$ on the number of distinct prime factors of $q$, from (1.4) it follows immediately that $\omega(q)\ll y^{1/2}/\log y$ (the implied constant is independent of $A$ if (say) $Q_{A}\geqslant A^{4A}$). Therefore, $q$ satisfies one of the conditions $(C_{1})$ or $(C_{2})$ of [4, Corollaire 2.2], and so an application of [4, Théorème 2.1] allows us to conclude that

provided both quantities $y$ and $u$ exceed a certain absolute constant; note that the implied constants in (2.5) are absolute. Combining (2.5) with [12, Theorem 1] it follows that

Since $0<\alpha\ll 1$ the quantities $\Psi(\tfrac{1}{2}x,y;\chi_{0})$ and $\Psi(2x,y;\chi_{0})$ are comparable in size; thus, as $\mathbf{1}_{[0,\frac{1}{2}]}\leqslant\Phi\leqslant\mathbf{1}_{[0,2]}$ one finds that

Following [10, 22] we now denote

with

Using (2.3) (with $c\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=\alpha$), the sum on the left side of (2.2) satisfies the bound

For the specific moduli considered in Theorem 1.1 and in Theorem 1.4, we show that the $L$-functions $L(s,\chi)$ attached to characters $\chi$ modulo $q$ have no zeros close to one, with at most one exceptional “problem character” $\chi_{\bullet}$ (in the sense of [22]). More precisely, we need to know (see Proposition 2.1 in §2.3 below):

where $D$ is the absolute constant described in [10, Rodosskiĭ Bound 1]. This means that the “problem range” (in the sense of Harper [10]) can be handled easily (this is definitely not the case in situations where (2.7) fails). It is precisely for this reason that we have been able to remove the obstruction $A<4\sqrt{e}$ encountered in the previous papers [10, 22] in the case that $\mathcal{Q}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}=\mathbb{N}$.

Assume from now on that $\mathcal{Q}$ is a set of natural numbers such that (2.7) holds for every $q\in\mathcal{Q}$. Taking the above considerations into account, and introducing the notation

to verify (2.2) (and establish ${\tt SC}\llbracket\hskip 1.0pt\mathcal{Q},A\hskip 1.0pt\rrbracket$) it suffices to show that

holds for a problem character $\chi_{\bullet}\in\mathcal{A}$, and that

where $W$ is defined by (2.6). For the most part, the results we need are already contained in [10]; these are briefly reviewed in §2.3.

In what follows, we use the terminology (cf. [10, p. 184])

• •

  $y$ is “large” if $e^{(\log x)^{0.1}}<y\leqslant x$;

• •

  $y$ is “small” if $(\log\log x)^{3}\leqslant y\leqslant e^{(\log x)^{0.1}}$;

• •

  $y$ is “very small” if $y<(\log\log x)^{3}$.

We also say (cf. [10, p. 186]) that $k$ lies in

• •

  the “basic range” if $\sqrt{u}\leqslant k\leqslant\tfrac{1}{2}\log q$;

• •

  the “Rodosskiĭ range” if $k_{0}\leqslant k<\sqrt{u}$,

• •

  the “problem range” if $0\leqslant k<k_{0}$,

where $k_{0}\mathrel{\vbox{\hbox{\scriptsize.} \hbox{\scriptsize.}}}={\left\lceil 4A\log A+D\right\rceil}$ as in (2.7). Again, we emphasize that our proofs of Theorems 1.1 and 1.4 essentially amount to showing that the problem range contains at most one character $\chi_{\bullet}$ in each case.

Both Theorems 1.1 and 1.4 follow from the following general statement (which does not assume anything about the arithmetic structure of the modulus $q$).

We apply two familiar principles that are commonly used in treatments of Linnik’s Theorem. The first is the zero-free region for Dirichlet $L$-functions (see Gronwall [9], Landau [17] and Titchmarsh [23]).

The second principle, which is due to Linnik [18], is often referred to as the “Deuring-Heilbronn” phenomenon.

The proof of Theorem 1.1 relies heavily on the following result of Chang [3, Corollary 9], which bounds short character sums over intervals for certain primitive characters with a smooth conductor.

We apply the following corollary of Lemma 3.3, which provides a weaker bound but has the advantage that it can be applied to longer intervals.