Elsevier

Advances in Mathematics

Volume 370, 26 August 2020, 107175
Advances in Mathematics

Breaking the 12-barrier for the twisted second moment of Dirichlet L-functions

https://doi.org/10.1016/j.aim.2020.107175Get rights and content

Abstract

We study the second moment of Dirichlet L-functions to a large prime modulus q twisted by the square of an arbitrary Dirichlet polynomial. We break the 12-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than q51/101=q1/2+1/202. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet L-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

Introduction

We study the mean square of the product of Dirichlet L-functions with arbitrary Dirichlet polynomials. The central problem is to obtain an asymptotic formula forχ(modq)|L(12,χ)|2|aqκαaχ(a)a|2, where denotes summation over all primitive characters χ modulo q, the coefficients αaaε are arbitrary, and 0<κ<1. The asymptotic evaluation when κ<1/2 was established by Iwaniec and Sarnak [14]. In this regime the main term comes from the “diagonal” contribution, and the off-diagonal terms contribute only to the error.

In this work we obtain an asymptotic expression for (1.1) with κ going beyond the 12-barrier. In this larger regime some off-diagonal terms contribute to the main term, and evaluating this contribution, as well as bounding the error terms, is considerably more difficult. Duke, Friedlander and Iwaniec [5] proved that the quantity in (1.1) may be bounded by Oε(q1+ε) for some κ>1/2, but their proof does not extend to give an asymptotic formula. Very recently, Conrey, Iwaniec and Soundararajan [4] applied their asymptotic large sieve and studied (1.1) with an additional averaging over the modulus q.

The analogous problem to (1.1) for the Riemann zeta-function was studied by Bettin, Chandee and Radziwiłł [2], who broke the 12-barrier for an arbitrary Dirichlet polynomial. Our work is inspired by their beautiful paper, but there are significant differences between the family of primitive Dirichlet L-functions in the q-aspect and the Riemann zeta-function in the t-aspect. These differences usually make the family of Dirichlet L-functions more difficult to work with. In fact, Bettin, Chandee and Radziwiłł mentioned the work of Duke, Friedlander and Iwaniec [5], and said: “Our proof of Theorem 1 would not extend to give an asymptotic in this case, and additional input is needed.”

It turns out that it is more convenient to work with a more general version of (1.1) by introducing “shifts”. We let α,βC satisfy Re(α),Re(β)(logq)1 and, for our later application2, Im(α),Im(β)logq. Furthermore, we treat the characters χ according to their parity to ensure that the L-functions under consideration have the same gamma factors in their functional equations. Let φ+(q) be the number of even primitive characters χ modulo q, and let + denote summation over all these characters. In this work we deal exclusively with even Dirichlet characters, but our arguments go through identically for odd characters. For fixed 0<κ<1, we studyIα,β=1φ+(q)+χ(modq)L(12+α,χ)L(12+β,χ)|A(χ)|2, whereA(χ)=aqκαaχ(a)a, and αa is an arbitrary sequence of complex numbers satisfying αaεaε.

For technical convenience we assume that q is prime throughout the paper. Note that in this case, the number of primitive characters is φ(q)=q2, and we have φ+(q)=(q3)/2. It is likely that our methods could be adapted to the case of general q with more effort.

The following is our main theorem.

Theorem 1.1

Suppose that q is prime. Let α,βC satisfy|Re(α)|,|Re(β)|(logq)1and|Im(α)|,|Im(β)|logq. Suppose that κ<1/2+1/202. ThenIα,β=ζ(1+α+β)da,dbqκ(a,b)=1αdaαdbda1+βb1+α+(qπ)(α+β)Γ(1/2α2)Γ(1/2β2)Γ(1/2+α2)Γ(1/2+β2)ζ(1αβ)da,dbqκ(a,b)=1αdaαdbda1αb1β+O(qδ0) for some δ0>0.

The range of κ can be enlarged if we know more about the Dirichlet polynomial A(χ). Let γ be a smooth function supported in [1,2] such that γ(j)jqε for any fixed j0. Suppose that α=ηλ, where ηa1,λa2 are two sequences of complex numbers supported on [1,A1] and [1,A2], respectively, with A1=qκ1, A2=qκ2 and κ=κ1+κ2, and satisfy ηa,λaaε. Friedlander and Iwaniec [7] showed that1φ(q)χ(modq)|nχ(n)nγ(nN)|2|A(χ)|2εqε+q3/4+5κ/4+ε(A1+A2)1/4, if Nq1/2+ε. When A1A2, their result gives an upper bound of Oε(qε) provided that κ<1/2+1/22.

Theorem 1.2

Assume as above and suppose further that 9κ+max{κ1,κ2}<5. Then (1.2) holds for some δ0>0.

If κ1=κ2, then Theorem 1.2 allows us to take κ<1/2+1/38.

Another case of special interest is when α=ηλ with ηa being smooth coefficients up to q1/2+ε and λa being arbitrary and as long as possible. This can be viewed as an analogue of Hough's result [12, Theorem 4] (see also [23, Theorem 1.1]), which gives an asymptotic formula for the fourth moment of Dirichlet L-functions twisted by the square of a Dirichlet polynomial of length less than q1/32.

Suppose thatηa1=η(a1A1), where η is a smooth function supported in [1,2] such that η(j)jqε for any fixed j0. If N,A1q1/2+ε, then Watt [21] proved that1φ(q)χ(modq)|nχ(n)nγ(nN)|2|A(χ)|2εqε+qϑ1/2+εA22, where ϑ=7/64. This yields an upper bound of Oε(qε) provided that κ2<1/4ϑ/2.

Theorem 1.3

Assume as above and suppose further that κ2<1/14ϑ/7 with ϑ=7/64. Then (1.2) holds for some δ0>0.

As an application of Theorem 1.1, we obtain the order of magnitude of the third moment of Dirichlet L-functions.

Theorem 1.4

Suppose that q is prime. Then1φ(q)χ(modq)|L(12,χ)|3(logq)9/4.

We remark that it is also possible to obtain upper bounds on all moments below the fourth by adapting the work of Radziwiłł and Soundararajan [18] and Hough [12] (see also [9]).

An important application of the twisted second moment of Dirichlet L-functions is in regard to non-vanishing of Dirichlet L-functions at the central point s=1/2. It is widely believed that L(1/2,χ)0 for all primitive characters χ. At least 34% of Dirichlet L-functions in the family of primitive characters, to a large modulus q, are known to not vanish at the central point [3] (see also [14]). Our results here, together with the mollifier method, may be used to give a slight improvement of such a result. However, we note that in the case of large prime q, Khan and Ngo [15], using the “twisted” mollifier introduced by [20] and [16], have obtained a non-vanishing proportion of 3/8. Improving that using the usual mollifier would require a mollifier with κ>3/5ϵ, which seems out of reach of the current techniques.

The use of Theorem 2 of [6] and our Proposition 3.2 below also suffices to break the 12-barrier. With those results, Theorem 1.1 holds provided that κ<1/2+1/526.

We close the introduction with a brief discussion of the main differences between our work and the work of Bettin, Chandee and Radziwiłł's on the twisted second moment of the Riemann zeta-function [2].

In [2], after applying the approximate functional equation and dyadic decompositions, the main object to study is of the form1ABMNaA,bBmM,nN,MNT1+εαaβbWˆ(Tlogambn), where W, say, is some compactly supported smooth function satisfying W(j)jTε for any fixed j0. The diagonal contribution am=bn may be extracted and is fairly easy to understand. For the remaining terms, write ambn=r with r0. The rapid decay of the Fourier transform implies that the contribution of the terms with |r|>T1+εABMN is negligible. The off-diagonal contribution is then essentially1ABMN0<|r|T1+εABMNambn=raA,bBmM,nN,MNT1+εαaβb. In particular, this is null unless AMBN. Writing ambn=r as a congruence condition modulo b and applying the Poisson summation formula transform the above expression to an exponential sum roughly of the formMNAB(AM+BN)0<|r|T1+εABMN|g|Tε(AM+BN)/MNaA,bBαaβbe(rgab). Trivially, this is bounded by T1+εABT2κ1+ε. So any extra power saving is sufficient to break the 12-barrier. Bettin, Chandee and Radziwiłł [2] gained this by utilizing an estimate for sums of Kloosterman fractions in [1] (or, [6]).

In our situation, we use the approximate functional equation and apply character orthogonality, as well as dyadic decompositions, to reduce the problem to understanding sums of the form1ABMNam±bn(modq)aA,bBmM,nN,MNq1+εαaβb. Diagonal main terms arise from am=bn. We set this contribution aside, and study the remaining terms. At this point, we can, as above, write the congruence condition modulo q as ambn=qr with r0, and switch to a congruence condition modulo b. The Poisson summation formula then leads to an exponential sum roughly of the formMNAB(AM+BN)0<|r|q1(AM+BN)|g|qε(AM+BN)/MNaA,bBαaβbe(qrgab).

The results on cancellation in sums of Kloosterman fractions [6], [1] are still applicable, but only work in the “balanced” case when AM and BN are more or less of the same size. Roughly speaking, the t-aspect averaging yields the constraint AMBN and ensures that r,gTκ1/2+ε in (1.4). In our situation (1.5), we lack the condition AMBN, and therefore the ranges of summation of r and g can be as large as qκ+ε. The trivial bound, which can be Oε(q2κ1/2+ε), is worse in our case as well. So a different method is required for the “unbalanced” regime when AM and BN are of rather different sizes. Furthermore, it is impossible to ignore the contribution of these terms. This contribution is genuinely large, and, as it turns out, will cancel out with the contribution from the principal character modulo q. A similar phenomenon already arose in Young's work on the fourth moment of Dirichlet L-functions [22]. We remark that it is possible to show that Iα,βεqε for some κ>1/2 without considering the unbalanced case [5].

We begin the treatment of the unbalanced regime by applying the Poisson summation formula to introduce exponential phases. The zero frequency cancels out with the contribution from the principal character modulo q. We bound the contribution of the non-zero frequencies with a delicate argument involving the additive large sieve inequality. The phases of the exponentials are rational fractions, and we divide these fractions into two classes: “good” fractions and “bad” fractions. The good fractions are far apart from each other, and we can immediately apply the additive large sieve inequality to get a saving. The bad fractions can be close together, which weakens the large sieve inequality, but we still obtain a saving since there are comparatively few of these bad fractions.

Lastly, we remark that with our two different methods for the two different regimes, the critical ranges of summation in (1.5) are when ABNqκ and Mq23κ. This explains why the ranges of κ for the asymptotic formula in our theorems are slightly smaller than the corresponding ranges for the upper bound.

Remark 1.5

Throughout the paper ε denotes an arbitrarily small positive number whose value may change from one line to the next.

Section snippets

Initial manipulations

We start by recalling the orthogonality property of characters and the approximate functional equation.

Lemma 2.1 Orthogonality

For (mn,q)=1 we have+χ(modq)χ(m)χ(n)=12d|qd|(m±n)μ(qd)φ(d)=12q|(m±n)φ(q)1.

Proof

The proof is standard. See, for example, [14, (3.1) and (3.2)]. 

Lemma 2.2 Approximate functional equation

Let χ be an even primitive character and let G(s) be an even entire function of rapid decay in any fixed strip |Re(s)|C satisfying G(0)=1. LetX±(s)=G(s)Γ(1/2±α+s2)Γ(1/2±β+s2)Γ(1/2+α2)Γ(1/2+β2) andV±(x)=12πi(ε)X±(s)xsdss. ThenL(12+α,χ)L(12+β,χ)=m,n

Main propositions

In this section we focus on the sumam±bn(modq)ambnαaβb over dyadic intervals. Theorem 1.1 is deduced from the following two propositions. The first result essentially treats the case when m and n are close, while the second one deals with the case when m and n are far apart.

Proposition 3.1

Let A,B,M,N1, and let αa,βb be two sequences of complex numbers supported on [A,2A] and [B,2B] satisfying αaAε,βbBε. Let W1 and W2 be smooth functions supported in [1,2] such that W1(j),W2(j)jqε for any fixed j0. LetS

Proof of Theorem 1.1

We proceed from (2.2) and start the evaluation of the off-diagonal terms, Sα,β+. We assume that the sequences αa,βb are supported on [A,2A] and [B,2B]. We remark that our main term analysis is inspired by the nice papers of Young [22] and Zacharias [23].

Proof of Theorem 1.2

We argue the same as in the proof of Theorem 1.1. The only difference is that we also apply Proposition 3.3 to (4.1): if (M,N)A1 we apply (3.1) of Proposition 3.1; if (M,N)A2,<A2,> we apply Proposition 3.2; and in the remaining case we apply Proposition 3.3. So it remains to check that the error term E(M,N) is acceptable when (M,N)A3.

As in Subsection 4.2, if (M,N)A3, then we may assume that M,Nqκ+6δ0. Without loss of generality, let us assume that AMBN. Then from Proposition 3.3,E(M,N)εq

Proof of Theorem 1.4

The lower bound for (1.3) is relatively straightforward, and follows from the work of Rudnick and Soundararajan [19]. We therefore focus on the upper bound. Our approach utilizes a combination of ideas from Heath-Brown [10] and Bettin, Chandee and Radziwiłł [2], as well as our Theorem 1.1 on the twisted second moment of Dirichlet L-functions. Heath-Brown [10, Theorem 1] previously obtained Theorem 1.4 assuming the Generalized Riemann Hypothesis, and Bettin, Chandee and Radziwiłł [2, Corollary 2]

Acknowledgments

The second author was supported by NSF grant DMS-1501982. The authors would like to thank Sandro Bettin and Maksym Radziwiłł for various helpful comments. The authors would like to thank the referees for pointing out very useful clarifications and insights that have increased the quality of the manuscript.

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    Current address: All Souls College, University of Oxford, UK.

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