Bilinear forms in Weyl sums for modular square roots and applications

Abstract

Let q be a prime, $P⩾1$ and let $N_q(P)$ denote the number of rational primes $p⩽P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\epsilon }⩽P⩽q$, for any fixed $\epsilon >0$. This improves upon what is implied by work of Pollack and Benli–Pollack.

The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salié sums). Our estimate has a power saving in the so-called Pólya–Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in ${\mathbb{F}}_q$.

This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted L-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdős–Odlyzko–Sárközy.

Introduction

Our motivation begins in the early 20th century. This is when I. M. Vinogradov initiated the study of the distribution of both quadratic residues and non-residues modulo a prime q. This remains a central theme in classical analytic number theory.

Several fundamental conjectures are still unresolved. Vinogradov's least quadratic non-residue conjecture asserts that

$$n_q⩽q^{o(1)},$$

where $n_q$ denotes the least non-quadratic residue modulo q ($n_q$ is necessarily prime). The best unconditional bounds known are

$$n_q⩽q^{1/(4\sqrt{e})+o(1)},$$

largely due to Burgess' bounds for character sums [11] and ideas of Vinogradov.

It is also natural to consider the least prime quadratic residue $r_q$ modulo q. The best unconditional result is due to Linnik and Vinogradov [53], where they showed

$$r_q⩽q^{1/4+o(1)}.$$

Much of the above discussion can be generalised to prime residues and non-residues of an arbitrary Dirichlet character χ of order k. One can see the work of Norton [43] for the analogue of (1.1) and of Elliot [21] for the analogue of (1.2).

Given the existence of a small prime quadratic residue and non-residue, the next natural question to ask is how many such quadratic residues and non-residues exist in a given interval $[2,P]$. Making use of reciprocity relations and the sieve, Benli and Pollack [4] have results in this direction for quadratic, cubic and biquadratic residues. For general Dirichlet characters one can see work of Pollack [44] and also a more recent work of Benli [3]. The impact and links of results of this type stretch far beyond analytic number theory. For example, Bourgain and Lindenstrauss [10, Theorem 5.1] are motivated by links with the Arithmetic Quantum Unique Ergodicity Conjecture. They have shown that for any $\epsilon >0$, there exists some $\delta >0$, such that for any sufficiently large D, the set

$$\mathcal{R}:=\{p\text{prime}:D^{\delta }⩽p⩽D^{1/4+\epsilon },\left(\frac{D}{p}\right)=-1\}$$

satisfies

$$\sum _{p\in \mathcal{R}}\frac{1}{p}⩾\frac{1}{2}-\epsilon .$$

When one relaxes the condition that a non-residue be prime, one does much better, as in the case of Banks, Garaev, Heath-Brown and Shparlinski [1], who show that for each $\epsilon >0$ and $N⩾q^{1/4\sqrt{e}+\epsilon }$, the proportion of quadratic non-residues modulo q in the interval $[1,N]$ is bounded away from 0 when q is large enough.

Furthermore, if $q\equiv 3(\mathrm{mod}4)$ is a large prime, then quadratic reciprocity tells us that the following two Legendre symbols are equal: $(-q/p)=(p/q)$. Thus counting quadratic residues modulo q is equivalent to counting small rational primes $p⩽P$ that split in the imaginary quadratic field $F:=\mathbb{Q}(\sqrt{-q})$. In particular, throughout this paper,

$$\chi (\cdot ):=\left(\frac{-q}{\cdot }\right)$$

defined via the Jacobi symbol, always denotes the character attached to F, and $L(s,\mathit{\chi })$ denotes the Dirichlet L-function attached to χ. Let $N_q(P)$ denote the number of rational primes $p⩽P$ that split in F.

A good reference point for the strength of our results is what the Generalised Riemann Hypothesis (GRH) implies. It is well known that under the GRH for $L(s,\chi )$ that we have $N_q(P)⩾c_{1}P/\log P$ for $P⩾c_2{(\log q)}^2$ for some absolute constants $c_1,c_2>0$, see, for example, [42, Section 13.1, Excercise 5(a)].

Furthermore, a result of Heath-Brown [33, Theorem 1] immediately implies the following. For any fixed $\epsilon >0$, for all but $o(Q/\log Q)$ primes $q⩽[Q,2Q]$, we have $N_q(P)=(1/2+o(1))P/\log P$ for $P⩾q^{\epsilon }$, as $Q\to \infty$.

The second theme which we develop in this paper concerns bounds of certain bilinear sums closely related to correlations between values of Salié sums

$$S(m,n;q)=\sum _{x\in {\mathbb{F}}_q}\left(\frac{x}{q}\right){\mathbf{e}}_q(mx+n\bar{x}),$$

see [45]. We emphasise that this is closely related to recent work of the first and fourth authors [19], who have computed a second moment for L-functions attached to a half-integral weight Kohnen newform, averaged over all primitive characters modulo a prime. Power savings in the error term for such a moment come in part from savings in the bound on correlations between Salié sums (1.3). Our argument gives a direct improvement of [19, Theorem 1.2], see Appendix C. It remains to investigate whether this improvement propagates into a quantitative improvement in the error term of [19, Theorem 1.1] for the second moment of the above L-functions.

Let $M,N$ be two positive real numbers and $\mathit{\alpha }={({\alpha }_m)}_{m\sim M}$ and $\mathit{\beta }={({\beta }_n)}_{n\sim N}$ be complex weights supported on dyadic intervals $m\sim M$ and $n\sim N$, where $a\sim A$ indicates $a\in [A,2A)$. Let $K:{\mathbb{F}}_q\to \mathbb{C}$ be some function, usually called a kernel. A bilinear form involving K is a sum of the shape

$$\sum _{m\sim M}\sum _{n\sim N}{\alpha }_m{\alpha }_{n}K(mn).$$

Bounds of such sums also have key automorphic and arithmetic applications.

In a series of two recent breakthrough papers using deep algebro–geometric techniques, Kowalski, Michel and Sawin [37], [38] have established non-trivial estimates for bilinear forms with Kloosterman sums in [37] and generalised Kloosterman sums in [38]. In particular, their estimates apply below the Pólya–Vinogradov range, that is, when the ranges of summation are $M,N\sim q^{1/2}$; this is where completion and Fourier theoretic methods breakdown. Such bounds are a crucial ingredient to the evaluation of asymptotic moments of ${\mathrm{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ L-functions over primitive Dirichlet characters, with power saving error (as well as many more automorphic applications), see [7], [37], [38], [55] and references therein.

It is important to note that the bilinear sums we study here do not fall under the umbrella of the results of [38]. The initial approach of [38] (Vinogradov's ab shifting trick and the Riemann Hypothesis for algebraic curves over a finite field) can be used. However, our approach leads to a much stronger result, for comparison see Appendix B. For an alternative treatment of bilinear forms in classical Kloosterman sums using the sum-product phenomenon, see [47].

The above two themes:

• lower bounds on the number of prime quadratic residues and non-residues;

• bounds of bilinear sums with modular square roots;

come together in the third direction which we pursue here: the distribution of square roots modulo q of primes $p⩽P$. Indeed, in the asymptotic formula we obtain, the main term is controlled by the counting function of prime quadratic residues, while the error term is given by the discrepancy, and depends on the quality of our bounds for certain bilinear sums we estimate in this paper.

We remark that it is natural to attempt to improve the error term on the average square of some L-functions from [19]. This however requires substantial effort with optimisation and balancing a number of estimates and so falls outside of the scope of this paper.

We are now ready to state some of the results in this paper. A high level sketch of the ideas and methodology in the proofs is deferred to Section 2.

Our first result is an unconditional lower bound for $N_q(P)$, but with ineffective constant.

Theorem 1.1

For any fixed $\epsilon >0$, any sufficiently large prime q, and any P with $q⩾P⩾q^{1/4+\epsilon }$, we have

$$N_q(P)⩾c(\epsilon )\min \{P^{1/2}{q}^{-\epsilon /2},P{q}^{-1/4-2\epsilon /3}\},$$

where $c(\epsilon )>0$ depends only on ε.

Remark 1.2

As with the results on quadratic residues of Benli and Pollack [3], [4], [44], Siegel's theorem is used in the proof of Theorem 1.1, and so the constant $c(\epsilon )$ is ineffective.

Observe that for $\epsilon >0$ small and $A>0$ fixed [44, Theorem 1.3] guarantees that

$$N_q(q^{1/4+\epsilon })⩾c(\epsilon ,A){(\log q)}^A,$$

for some constant $c(\epsilon ,A)>0$ that depends only on ε and A. Theorem 1.1 above improves this to a small power of q, that is,

$$N_q\left(q^{1/4+\epsilon }\right)⩾c(\epsilon )q^{\epsilon /3}.$$

This was also proved independently by Benli [3] (taken with $k=2$) very recently. Theorem 1.1 also substantially improves the lower bound

$$N_q(P)⩾P^{1/25}{q}^{-1/50},q^{1/2+\epsilon }⩽P⩽q,$$

established by Benli and Pollack [4, Theorem 3] and in particular, implies

$$N\left(q^{1/2+\epsilon }\right)⩾q^{1/4},$$

for any $\epsilon >0$, provided that q is large enough.

Our next result is an unconditional bound for $N_q:=N_q(q)$, with effective constant.

Theorem 1.3

Suppose $q⩾67$ is prime with $q\equiv 3(\mathrm{mod}16)$. Then

$$N_q>\frac{(2-\log (3\sqrt{2}))}{2}\frac{\lfloor \sqrt{3q/4}\rfloor }{\log q}.$$

Since

$$\frac{(2-\log (3\sqrt{2}))\sqrt{3}}{4}=0.2402\dots ,$$

we see from Theorem 1.3 that there is an effectively computable absolute constant $c_0$ such that for $q⩾c_0$ we have

$$N_q>\frac{0.24\sqrt{q}}{\log q}.$$

In fact one can get a better constant by estimating certain quantities more carefully.

Remark 1.4

Unfortunately the argument of the proof of Theorem 1.3 does not scale to estimate $N_q(P)$ with $P<q$. However it actually increases its strength for $P>q$, which is still a meaningful range in the problem of estimating $N_q(P)$. We do not consider this case as small values of P are of our principal interest.

Many famous unsolved conjectures are known to hold under the assumption of a Siegel zero [26]. This is because one can sometimes break the parity problem of the sieve with this hypothesis. A notable example is Heath-Brown's proof [32] of the twin prime conjecture assuming Siegel zeros. Continuing this tradition, we prove an essentially sharp lower bound for $N_q(P)$ under the assumption that $L(s,\chi )$ has a mild Siegel zero.

Theorem 1.5

Suppose $q\equiv 3(\mathrm{mod}4)$ is a large prime and

$$L(1,\chi )=O(1/{(\log q)}^{10}).$$

Then for any fixed $\epsilon >0$ and P with $q^{1/2+\epsilon }⩽P⩽q$, we have

$$N_q(P)⩾c(\epsilon )h(-q)\frac{P}{\sqrt{q}{(\log q)}^2},$$

where $h(-q)$ is the class number of $F=\mathbb{Q}(\sqrt{-q})$ and $c(\epsilon )>0$ depends only on ε.

Remark 1.6

The constant $c(\epsilon )$ in Theorem 1.5 is ineffective.

Counting small split primes in general number fields is a notoriously difficult and fundamental problem. Ellenberg and Venkatesh [20] have established a direct connection between counts for small split primes and bounds for ℓ-torsion in general class groups.

We also refer to [28], [29], [30] for results and references on lower bounds on class numbers of imaginary quadratic fields.

We recall that $a\sim A$ means $a\in [A,2A)$. Given two real numbers $M,N$ and complex weights

$$\mathit{\alpha }={({\alpha }_m)}_{m\sim M}\text{and}\mathit{\beta }={({\beta }_n)}_{n\sim N},$$

we denote

$${\Vert \mathit{\alpha }\Vert }_{\infty }:=\underset{m\sim M}{\max }|{\alpha }_m|\text{and}{\Vert \mathit{\alpha }\Vert }_{\sigma }:={(\sum _{m\sim M}|{\alpha }_m{|}^{\sigma })}^{\frac{1}{\sigma }},$$

and similarly for β.

For $a,h\in {\mathbb{F}}_q^{\times }$ we consider bilinear forms in Weyl sums for square roots

$$W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N):=\sum _{m\sim M}\sum _{n\sim N}{\alpha }_m{\beta }_n\sum _{\begin{array}{c}\hfill x\in {\mathbb{F}}_q\hfill \\ \hfill x^2=amn\hfill \end{array}}{\mathbf{e}}_q(hx).$$

We notice that this is equivalent to studying bilinear forms with Salié sums, given by (1.3), thanks to the evaluation in [45], see also [35, Lemma 12.4] and [46, Lemma 4.4]:

$$\frac{1}{\sqrt{q}}S(m,n;q)=\frac{1}{\sqrt{q}}S(1,mn;q)={\epsilon }_q\left(\frac{n}{q}\right)\sum _{\begin{array}{c}\hfill x\in {\mathbb{F}}_q\hfill \\ \hfill x^2=mn\hfill \end{array}}{\mathbf{e}}_q\left(2x\right),$$

where ${\epsilon }_q=1$ if $q\equiv 1(\mathrm{mod}4)$ and ${\epsilon }_q=i$ if $q\equiv 3(\mathrm{mod}4)$ (note that if $(mn/q)=-1$, the Salié sum vanishes). The tuple of characters $(1,(\cdot /q))$ is Kummer induced (cf. [38, Section 2]), and so the results of [38] do not apply to (1.6).

Our goal is to improve the trivial bound

$$W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)=O\left({\Vert \mathit{\alpha }\Vert }_1{\Vert \mathit{\beta }\Vert }_1\right),$$

in the Pólya–Vinogradov range. In arithmetic applications the case when weights satisfy

$${\Vert \mathit{\alpha }\Vert }_{\infty },{\Vert \mathit{\beta }\Vert }_{\infty }=q^{o(1)}$$

is most important. However, our bounds are valid for more general weights. Making use of weighted additive energies, and some ideas from [9] and [12], we prove the following result.

Theorem 1.7

For any positive integers $M,N⩽q/2$ and any weights α and β as in (1.5), we have

$$\begin{gathered} |W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)|⩽{\Vert \mathit{\alpha }\Vert }_2{\Vert \mathit{\beta }\Vert }_{\infty }^{1/3}{\Vert \mathit{\beta }\Vert }_1^{2/3}{q}^{1/8+o(1)}{M}^{7/24}{N}^{1/8} \\ (\frac{M^{7/48}}{q^{1/16}}+1)(\frac{N^{7/48}}{q^{1/16}}+1) \end{gathered}$$

and

$$|W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)|⩽{\Vert \mathit{\alpha }\Vert }_2{\Vert \mathit{\beta }\Vert }_1^{3/4}{\Vert \mathit{\beta }\Vert }_{\infty }^{1/4}{q}^{1/8+o(1)}{M}^{5/16}{N}^{1/16}(\frac{M^{3/16}}{q^{1/8}}+1)(\frac{N^{3/16}}{q^{1/8}}+1).$$

When the weights α and β satisfy (1.8), we can re-write the bounds of Theorem 1.7 in the following simplified form

$$\begin{gathered} |W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)|⩽q^{1/8+o(1)}{(MN)}^{19/24} \\ (\frac{M^{7/48}}{q^{1/16}}+1)(\frac{N^{7/48}}{q^{1/16}}+1) \end{gathered}$$

and

$$\begin{gathered} |W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)|⩽q^{1/8+o(1)}{(MN)}^{13/16} \\ (\frac{M^{3/16}}{q^{1/8}}+1)(\frac{N^{3/16}}{q^{1/8}}+1). \end{gathered}$$

Power savings in the error term of an asymptotic formula for a second moment of certain L-functions [19] comes from savings in the bound on $W_{a,q}(\mathit{\alpha },\mathit{\beta };h,M,N)$ in the Pólya–Vinogradov range, as well as other ranges in which spectral techniques are used. Additional ideas are also needed in [19] to make the asymptotic formula unconditional, because ${\alpha }_m$ and ${\beta }_n$ are coefficients of a fixed normalised Kohnen newform. It is not known yet that they satisfy (1.8) (in this context, the condition (1.8) is the Ramanujan–Petersson conjecture, also equivalent to the Lindelöf hypothesis for the twisted L-function attached to the Shimura lift). The best known bound is ${\alpha }_m=O\left(m^{1/6+\epsilon }\right)$ due to Conrey and Iwaniec [13].

We now outline the arithmetic applications of Theorem 1.7.

We recall that the discrepancy $D_N$ of a sequence in ${\xi }_1,\dots ,{\xi }_N\in [0,1)$ is defined as

$$D_N=\underset{0⩽\alpha <\beta ⩽1}{\sup }|\mathrm{\#}\{1⩽n⩽N:{\xi }_n\in [\alpha ,\beta )\}-(\beta -\alpha )N|,$$

where $\mathrm{\#}\mathcal{S}$ denotes the cardinality of $\mathcal{S}$ (if it is finite), see [15], [39] for background. For positive integers P and R, denote the discrepancy of the sequence (multiset) of points

$$\{x/q:x^2\equiv pr(\mathrm{mod}q)\text{for some primes}p⩽P,r⩽R\}$$

by ${\mathrm{\Delta }}_q(P,R)$. Combining the bound (1.9) with the classical Erdős–Turán inequality, see Lemma 3.1 below, we derive an equidistribution of the modular square roots of products of two primes.

Corollary 1.8

For $1⩽P,R⩽q$ we have

$${\mathit{\Delta }}_q(P,R)⩽q^{1/8}{(PR)}^{19/24+o(1)}(\frac{P^{7/48}}{q^{1/16}}+1)(\frac{R^{7/48}}{q^{1/16}}+1).$$

Our next application is to a relaxed version of the still open problem of Erdős, Odlyzko and Sárközy [22] on the representation of all reduced classes modulo an integer m as the products pr of two primes $p,r⩽m$. This question has turned out to be too hard even for the Generalised Riemann Hypothesis, thus various relaxations have been considered, see [48] for a short overview of currently available results in this direction.

Here we obtain the following variant about products of two small primes and a small square. To simplify the result we assume that $p,r⩽q^{2/3}$. In this case, using the bound (1.10) one easily derives similarly to Corollary 1.8 the following result.

Corollary 1.9

Let $P,R⩽q^{2/3}$ be real numbers such that the interval $[2,P]$ contains $P^{1+o(1)}$ of both prime quadratic residues and non-residues. If $q⩾S⩾1$ is a real number such that

$${(PR)}^{3/16}S⩾q^{9/8+\epsilon },$$

then any reduced residue class modulo q can be represented as $pr{s}^2$ for two primes $p⩽P$, $r⩽R$ with some real $P,R⩽q^{2/3}$ and a positive integer $s⩽S$.

In particular, if $P=R=S$ then the result of Corollary 1.9 is nontrivial if $P⩾q^{9/11+\epsilon }$, while for $P=R=q$ we need $S⩾q^{3/4+\epsilon }$.

For a positive integer P we denote the discrepancy of the sequence (multiset) of points

$$\{x/q:x^2\equiv p(\mathrm{mod}q)\text{for some prime}p⩽P\}$$

by ${\mathrm{\Gamma }}_q(P)$.

Theorem 1.7 in combination with the classical Erdös–Turán inequality (see Lemma 3.1) and the Heath-Brown identity [31] yield the following result on the equidistribution of square roots of primes. Since we are mostly interested in the values of P which are as small as possible, we use only the first bound of Theorem 1.7. For large values of P one can get a better result but using the second bound as well or both, see also Remark 7.1 below.

Theorem 1.10

For any $P⩽q$ we have

$${\mathit{\Gamma }}_q(P)⩽q^{61/1760}{P}^{61/66+o(1)}+q^{13/110}{P}^{9/11+o(1)}.$$

If the interval $[2,P]$ contains $P^{1+o(1)}$ prime quadratic residues, which is certainly expected and is known under the GRH, see Sections 1.1 and 1.2, then Theorem 1.10 is nontrivial for $P⩾q^{13/20+\epsilon }$ with some fixed $\epsilon >0$.

We also point out that the equidistribution here is the opposite situation considered in [17], [18], [34], [41], [50], where the modulus is prime and varies.