Sums of singular series along arithmetic progressions and with smooth weights

The Hardy–Littlewood $k$-tuples conjectures, and the constants known as singular series that appear within them, have long been studied in connection to the distribution of primes. These conjectures state that for any $k$-tuple $\mathcal{H}=\{h_{1},\dots,h_{k}\}$ of distinct integers, the number of $k$-tuples of primes of the form $(n+h_{1},\dots,n+h_{k})$, with $n\leq x$, is given by

where $\mathfrak{S}(\mathcal{H})$ is the singular series

and $\nu_{\mathcal{H}}(p)$ denotes the number of distinct residue classes modulo $p$ occupied by the elements of $\mathcal{H}$.

In [1], Gallagher showed that the Hardy–Littlewood conjectures imply that the distribution of primes in intervals of size $h=\lambda\log x$ is Poissonian for fixed $\lambda$, by showing that the singular series is $1$ on average. In particular, he showed that for any fixed $k$,

In 2004, Montgomery and Soundararajan [7] used a more refined estimate of sums of singular series to show that when $h$ is in a larger regime, with $h/\log x\to\infty$ but $h=o(x)$, the Hardy–Littlewood conjectures imply that the distribution of primes, counted with von Mangoldt weights, becomes Gaussian with mean $\sim h$ and variance $\sim h\log\frac{x}{h}$, matching numerical data. Instead of the singular series itself, they considered alternating sums of singular series, defining $\mathfrak{S}_{0}(\mathcal{H}):=\sum_{\mathcal{J}\subset\mathcal{H}}(-1)^{|\mathcal{H}\setminus\mathcal{J}|}\mathfrak{S}(\mathcal{J}).$ These sums have the effect of subtracting the main term from the outset, making it easier to understand lower-order contributions. The analogous Hardy–Littlewood conjecture says that

so that each term on the left-hand side has expectation $0$.

Montgomery and Soundararajan [7] showed that for fixed $k$,

where $\mu_{k}=1\cdot 3\cdots(k-1)$ if $k$ is even, and $0$ if $k$ is odd, and $A=2-\gamma_{0}-\log 2\pi$, with $\gamma_{0}$ the Euler–Mascheroni constant. Their results depend crucially on work of Montgomery and Vaughan [8] on the distribution of reduced residues mod $q$ in short intervals.

Here we develop analogs of the work of Montgomery and Vaughan as well as that of Montgomery and Soundararajan by studying sums of singular series with added conditions on the set $\mathcal{H}$. First, instead of summing over all subsets of $[1,h]$, we restrict to sets whose elements lie in arithmetic progressions. For a fixed modulus $r$ and congruence classes $c_{1},\dots,c_{k}$ mod $r$, we study the sum

where $\mathfrak{S}_{0,r}(\mathcal{H})=\sum_{\mathcal{J}\subset\mathcal{H}}\mathfrak{S}_{r}(\mathcal{J})(-1)^{|\mathcal{H}\setminus\mathcal{J}|}$ and $\mathfrak{S}_{r}(\mathcal{H})$ is the singular series of $\mathcal{H}$ away from $r$, given by

The study of these sums is of interest for two reasons. First, they appear in the work of Lemke Oliver and Soundararajan in [5] on bias in the distribution of consecutive primes in arithmetic progressions. Specifically, Lemke Oliver and Soundararajan conjecture that if $\pi(x;q,(a,b))$ is defined as the number of primes $p\leq x$ with $p\equiv a\bmod q$ and $p_{\text{next}}\equiv b\bmod q$, then

where $\alpha(y):-1-\frac{q}{\phi(q)\log y}$, $\epsilon_{q}(a,b)$ is a constant defined only in terms of $q,a,$ and $b$, and

They proceed to heuristically estimate $\mathcal{D}(a,b;y)$, and thus $\pi(x;q,(a,b))$, by estimating a weighted version of $R_{2}(h;r,c_{1},c_{2})$. Our results generalize these arguments by estimating $R_{k}(h;r,c_{1},c_{2})$ for all $k$, which is necessary for understanding some of the error terms appearing in Lemke Oliver and Soundararajan’s heuristic. Secondly, restricting sums of singular series in this manner may shed light on other questions about sums of singular series. We show in Theorem 1.2 that the asymptotics for (4) are governed by incidences among the $c_{i}$’s mod $r$. As discussed in [4], we do not yet know the asymptotic average size of sums of $\mathfrak{S}_{0}(\mathcal{H})$ when $|\mathcal{H}|$ is odd. The results of these more refined averages of singular series may clarify where the main term should be coming from for sums of singular series with an odd number of terms.

The sums over $\mathfrak{S}_{0,r}$ admit the expansion

Following [7] and [8], we first consider a related quantity, where the summands $q_{i}$ are restricted to divide a secondary modulus $q>1$ and the $h_{i}$’s are not necessarily distinct, to get

where for $\alpha\in{\mathbb{R}}$,

In order to state our result, we will need to fix some notation concerning perfect matchings of $[1,k]$. For $k\geq 1$, a perfect matching $\sigma$ of $[1,k]$ is a set $\sigma=\{(i,j)\}$ of unordered pairs in $[1,k]$ so that each element is paired with exactly one other element, i.e. each $i$ appears in exactly one pair $(i,j)$, and $i\neq j$. Since each pair $(i,j)\in\sigma$ is unordered, we will generally choose to write the representative with $i<j$, so that $\sigma\{(i,j)\}$ with $i<j$. Let $\mathcal{B}_{k}$ denote the set of perfect matchings of $[1,k]$, so that

Note that when $k$ is odd, $\mathcal{B}_{k}=\varnothing$. Moreover, for a set of integers $\{a_{1},\dots,a_{k}\}$, we will denote by $\mathcal{B}(a_{1},\dots,a_{k})$ the set of matchings of $\{a_{1},\dots,a_{k}\}$ into pairs, so that $\mathcal{B}_{k}=\mathcal{B}([1,k])$.

In Section 2, we prove the following result, which mirrors Theorem 1 of [7].

In order to state our main result on the asymptotics of $R_{k}(h;r,c_{1},\dots,c_{k})$, we define some further notation. For $1\leq\ell\leq r$, define

Note that some of the sets $\mathcal{C}_{\ell}$ may be empty, and that $\bigcup_{\ell=1}^{r}\mathcal{C}_{\ell}=[1,k]$. We will say that a partition $\mathcal{P}=\{\mathcal{S}_{1},\dots,\mathcal{S}_{M}\}$ of $[1,k]$ refines $\{\mathcal{C}_{\ell}\}_{\ell\in[1,k]}$ if for each $\mathcal{S}_{m}\in\mathcal{P}$, there exists some $\ell$ with $\mathcal{S}_{m}\subset\mathcal{C}_{\ell}$; note that $\ell$ is then unique. For such a partition, write $\mathcal{P}\preceq\{\mathcal{C}_{\ell}\}_{\ell\in[1,k]}$ and define $c(\mathcal{S}_{m})$ to be the value $\ell$ with $\mathcal{S}_{m}\subset\mathcal{C}_{\ell}$.

The framework of Theorems 1.1 and 1.2 also applies to sums of singular series weighted by smooth functions. Let $f_{1},\dots,f_{k}:{\mathbb{R}}){\geq 0}\to{\mathbb{C}}$ be functions with compact supports contained in $(0,\infty)$ and such that $|\hat{f_{i}}(\xi)|\ll\xi^{-2}$, where the Fourier transform $\hat{f}$ is defined as

For $h\in{\mathbb{N}}$, we are interested in the quantity

The size of $R_{k}(h;f_{1},\dots,f_{k})$ as $h\to\infty$ is similar in structure to the analogous result for sums of singular series along arithmetic progressions: the main term is a sum over perfect matchings of $[1,k]$, and for each matching $\sigma$ the contribution is determined by interactions of $f_{i}$ and $f_{j}$ for each pair $(i,j)\in\sigma$.

We set some notation before stating our results. Define

where for $\alpha\in{\mathbb{R}}$ and $f$ smooth, with compact support, and such that $|\hat{f}(\xi)|=O(|\xi|^{-2})$,

Since $f$ is smooth, $E_{f,h}(\alpha)$ is a smoother indicator function of values of $\alpha$ that are close to $0$. In particular, by Poisson summation,

By assumption, $\hat{f}(\xi)=O(|\xi|^{-2})$. For any real number $\alpha$, only one value of $\alpha-n$ will be in the interval $[-1/2,1/2)$; let $\overline{\alpha}$ denote this value, so that $\overline{\alpha}$ is the representative of $\alpha\bmod 1$ satisfying $-1/2\leq\overline{\alpha}<1/2$. Then

The following results about $V_{k}(q,h;f_{1},\dots,f_{k})$ hold, analogous to Theorem 1.1 and Lemma 3.2.

Using precisely the same techniques as in the case of arithmetic progressions, an analog to Theorem 1.2 also holds for the smooth functions setting, where the input to each summand is clarified by Lemma 1.4.

The proofs of Theorems 1.3 and 1.5 are identical to the proofs of Theorems 1.1 and 1.2, so we omit them. However, Theorems 1.3 and 1.1, which provide asymptotics for $V_{k}$ in the cases of smooth weighting and arithmetic progressions, rely on lemmas about sums of $E_{f,h}(\alpha)$. The results are fundamentally the same, but the flavor of several of these lemmas is somewhat different in the smooth case, so we record them in Section 5. We also provide the proof of Lemma 1.4, which is similar to the proof of Lemma 3.2.

The organization of this paper is as follows. In Section 2, we prove Theorem 1.1. In Section 3, we compute certain sums of 2-term singular series which will then be used in the proof of Theorem 1.2. In Section 4, we prove Theorem 1.2. Finally, in Section 5 we discuss the smooth case.

The arguments in this section closely follow the arguments in the proof of Lemma 8 in [8] and Theorem 1 in [7]. Here we outline the argument; where it differs, we present detailed explanations, but many steps are cited to [8] and [7].

The functions $E_{r,c_{i}}(\alpha)$, defined in (7), are modular versions of sums $E(\alpha):=\sum_{m\leq h}e(m\alpha)$. The sums $E(\alpha)$ are large if $\alpha$ is close to an integer and otherwise exhibit large amounts of cancellation. Since $E_{r,c_{i}}(\alpha)$ is summed only over $m$ in a set congruence class modulo $r$, it will also take large values when $\alpha$ is close to a multiple of $r$. For any $c,r$, write

so that

Define

so that $|E_{r,c}(\alpha)|\leq F_{r}(\alpha).$ We can then closely follow the analysis of Montgomery and Vaughan in [8] and Montgomery and Soundararajan in [7]. In particular, $F_{r}(\alpha)$ can be bounded, up to some dependence on $r$, by the same bounds as appear in Lemmas 4,5,6, and 7 of [8] and using the same arguments. We arrive at the following result.

Theorem 2.1 accounts for all terms in $V_{k}(q,h;r,c_{1},\dots,c_{k})$ except for those terms where the $q_{i}$’s are equal in pairs and otherwise distinct. When $k$ is odd, there are no such terms, so assume that $k$ is even. By the same argument as in [7], we can drop the assumption that the pairs are distinct, so that the terms left to be estimated are given by

where for each $i$,

Each term $\sigma$ in our sum is identical up to changing labels, so without loss of generality we will work with the term $\sigma=\{(i,k/2+i):1\leq i\leq k/2\}$, so that $q_{i}=q_{i+k/2}$ for all $1\leq i\leq k/2$. This term is given by

Let $\mathcal{J}\subset[1,k/2]$ be the subset of $i$’s with $0<b_{i}<q_{i}$ instead of $b_{i}=q_{i}$. For $i\not\in\mathcal{J}$, we have

which, when summed over $1<q_{i}|q$ with the weight $\frac{\mu(q_{i})^{2}}{\phi(q_{i})^{2}}$, is equal to $V_{2}(q,h;r,c_{i},c_{k/2+i})$. Thus, (18) is equal to

where

The term $\mathcal{J}=\varnothing$ gives the desired main term, so it remains to show that the terms with $\mathcal{J}\neq\varnothing$ are smaller. By following the reasoning on page 11 of [7], we get that

and that for any $c_{1},c_{2}\bmod r$,

Applying these estimates to (19) completes the proof of Theorem 1.1.

In order to prove Theorem 1.2, we will ultimately invoke Theorem 1.1, which will then relate quantities involving $k$-term singular series to quantities involving $2$-term singular series. In this section, we state two lemmas which compute, respectively, sums of two-term singular series in arithmetic progressions, and the quantity $V_{2}(Q,h;r,c_{1},c_{2})$ when $Q$ is the product of all primes below $h^{2}$, which will be applied in Section 4.

The following lemma is a computation of sums of two-term singular series to the modulus $q$. Its proof is nearly identical to the proof of Proposition 2.1 in [5], so we omit it. Similar quantities were previously studied in [2] and [6].