Low-lying zeros of quadratic Dirichlet L-functions: A transition in the ratios conjecture

Low-lying zeros of quadratic Dirichlet L-functions: A transition in the ratios conjecture Abstract We study the 1-level density of low-lying zeros of quadratic Dirichlet L-functions by applying the L-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz–Sarnak heuristic as well as in the lower-order terms when the support of the Fourier transform of the corresponding test function reaches the point 1. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case. 1. Introduction In this paper, we study low-lying zeros in the family F*(X)≔{L(s,χ8d):1≤∣d∣≤X;disoddandsquarefree} of Dirichlet L-functions. Here we define the real primitive character of conductor 8∣d∣ by the Kronecker symbol χ8d(n)≔(8dn). Low-lying zeros for quadratic Dirichlet L-functions have been studied extensively in the literature, see, for example, Özlük and Snyder [17], Rubinstein [18], Gao [10], Entin et al. [6], Miller [15] and the authors’ previous paper [8]. The L-functions Ratios Conjecture of Conrey et al. [2, Section 5] is a recipe for obtaining conjectural formulas for averages of quotients of (products of) L-functions evaluated at certain values in the critical strip. This can in turn be used to give extremely precise predictions for a variety of statistics for families of L-functions. For example, this has been carried out notably by Conrey and Snaith [3, 4] for correlations of the zeros of the Riemann zeta function and the 1-level density and moments of L-functions, and more recently by Mason and Snaith [14] for the n-level density of orthogonal and symplectic families of L-functions. Our goal is to compare the Ratios Conjecture prediction of [3] with the estimate we obtained in [8] for the 1-level density in the family F*(X) of quadratic Dirichlet L-functions. While agreement between the leading term in the Ratios Conjecture and the Katz–Sarnak random matrix theory prediction [12] was shown previously in [13, 15], the novelty in our work is the isolation of a sharp transition in the lower-order terms that we give explicitly and which agrees with Rudnick’s work [19] over function fields and our previous results [8]. Before we describe the new results, we first review and refine our previous work. We begin by introducing the 1-level density of the family F*(X). Given a large positive number X, we set L≔log(X2πe) and W*(X)≔∑*doddw(dX), where w(t) is an even, non-zero and nonnegative Schwartz function and the star on the sum denotes a restriction to squarefree integers. We introduce the 1-level density of the family F*(X) as the functional D*(ϕ;X)≔1W*(X)∑*doddw(dX)∑γ8dϕ(γ8dL2π). (1.1) Here and throughout, ϕ will be a real and even Schwartz test function. Furthermore, we define γ8d≔−i(ρ8d−12), where ρ8d runs over the non-trivial zeros of L(s,χ8d) (that is, zeros with 0<R(ρ8d)<1). The Katz–Sarnak heuristic [12] provides a precise prediction of the statistics of low-lying zeros in families of L-functions (see also the recent paper [20]). In our situation, the heuristic asserts that F*(X) has symplectic symmetry type and in particular that limX→∞D*(ϕ;X)=ϕ^(0)−12∫−11ϕ^(u)du (1.2) independently of the support of ϕ^. Moreover, note that there is a phase transition in the right-hand side of (1.2) occurring when the supremum σ of the support of ϕ^ reaches 1. This transition has a strong influence on the shape of the lower order terms in D*(ϕ;X) (cf. [8, 19]) and will thus be of fundamental interest in the current paper. The lower order terms in the 1-level density for the family F*(X) were recently studied under the assumption of GRH in [8]. There we obtained an asymptotic formula for D*(ϕ;X) in descending powers of logX, which is valid when the support of ϕ^ is contained in (−2,2). In particular, we uncovered a phase transition when σ approaches 1 in the main term as well as in the lower order terms. This asymptotic formula (see [8, Theorem 3.5]) contains a term J(X) which is expressed in terms of explicit transforms of the weight function w. In the current paper, we give an asymptotic for this complicated expression (see Section 2), and, as a consequence, the following result holds. Theorem 1.1 Fix ε>0. Assume GRH and suppose that σ=sup(suppϕ^)<2. Then the 1-level density of low-lying zeros in the family F*(X)of quadratic Dirichlet L-functions whose conductor is an odd squarefree multiple of 8 is given by D*(ϕ;X)=ϕ^(0)+∫1∞ϕ^(u)du+ϕ^(0)L(log(2e1−γ)+2w^(0)∫0∞w(x)(logx)dx)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL)+J(X)+Oε(Xσ6−13+ε), (1.3)where J(X)is defined in (2.1) and satisfies the asymptotic relation J(X)=ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). (1.4) (Here Mwdenotes the Mellin transform of w.) Remark 1.2 Note that Theorem 1.1 is in agreement with the Katz–Sarnak heuristic (1.2) (cf. [8, Section 3]). In fact, in [8, Theorem 1.1], we show how to write (1.3) as D*(ϕ;X)=ϕ^(0)−12∫−11ϕ^(u)du+∑k=1KRw,k(ϕ)(logX)k+Ow,ϕ,K(1(logX)K+1), where Rw,k(ϕ) are linear functionals in ϕ that can be given explicitly in terms of w and the derivatives of ϕ^ at the points 0 and 1. An analogous expression having a transition at the point 1 was obtained by Rudnick [19, Corollary 3] in the function field case. In Conjecture 3.1, we adapt the Ratios Conjecture [3, Conjecture 2.6] for the family F*(X) to our situation (we need to add and keep track of the smooth weight function w). Following [3, Theorem 3.1], we use this conjecture to obtain the following asymptotic expression for D*(ϕ;X) (see Section 3). Theorem 1.3 Fix ε>0and let ϕbe an even Schwartz test function on Rwhose Fourier transform has compact support. Assume GRH and Conjecture3.1 (the Ratios Conjecture for F*(X)). Then the 1-level density for the low-lying zeros in the family F*(X)is given by D*(ϕ;X)=1W*(X)∑*doddw(dX)12π∫R(2ζ′(1+2it)ζ(1+2it)+2Aα(it,it)+log(8∣d∣π)+12Γ′Γ(14+a−it2)+12Γ′Γ(14+a+it2)−2Xd(12+it)ζ(1−2it)A(−it,it))·ϕ(tL2π)dt+Oε(X−12+ε),where A, Aα, aand Xdare defined by (3.8), (3.11), (3.4) and (3.10), respectively. Given the apparent difference between the formulas for D*(ϕ;X) in Theorems 1.1 and 1.3, it is an interesting question to ask whether the detailed information about the phase transition at σ=1 occurring in (1.3) is also present in Theorem 1.3. Our main theorem answers this question in the affirmative and, to the best of our knowledge, this is the first comprehensive investigation of such a transition via a Ratios Conjecture calculation (see Section 4). Theorem 1.4 Let ϕbe an even Schwartz test function on Rwhose Fourier transform has compact support. Assume GRH and Conjecture3.1 (the Ratios Conjecture for F*(X)). Then the 1-level density for the low-lying zeros in the family F*(X)can be written in the form D*(ϕ;X)=ϕ^(0)+∫1∞ϕ^(u)du+ϕ^(0)L(log(2e1−γ)+2w^(0)∫0∞w(x)(logx)dx)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL)+ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2).In particular, for test functions ϕwith σ=sup(suppϕ^)<2, this formula agrees with Theorem1.1up to an error term of order O(L−2). Remark 1.5 The error term in Theorem 1.4 results from a Taylor expansion of the integrand in (4.5) in powers of τL with remainder O(∣τ∣2L−2). In principle, with a higher order Taylor expansion, one can obtain an error term in Theorem 1.4 of size O(L−k) for any k≥2 by the same methods. However, for conciseness, we choose not to pursue this here. Remark 1.6 The first indications of a phase transition in a Ratios Conjecture calculation were given by Miller [15, Section 2.2]. However, the arguments in [15] contain several serious issues. In particular, the Katz–Sarnak main term was not computed correctly (cf. [15, Lemma 2.1, Lemma 2.6]). From the results in [8, 19], it is not clear whether or not we can expect the first lower order term to have the same shape also for test functions of larger support of their Fourier transforms. One of the consequences of Theorem 1.4 is that under the assumption of the Ratios Conjecture no additional phase transitions occur when the support of the Fourier transform reaches points in the interval [2,∞). 2. Proof of Theorem 1.1 As in [8] we define J(X)≔1L∫0∞(ϕ^(1+τL)eτ2∑n≥1h1(neτ2)+ϕ^(1−τL)∑n≥1h2(neτ2))dτ, (2.1) where h1(x)≔3ζ(2)w^(0)∑s≥1soddμ(s)s(g^(2sx)−g^(sx)),h2(x)≔3ζ(2)w^(0)∑s≥1soddμ(s)s2(12g(x2s)−g(xs)), and g(y)≔w^(4πey2). The formula (1.3) for D*(ϕ;X) was obtained in [8, Theorem 3.5]. Thus, it remains to prove the non-trivial estimate (1.4) for J(X). Using Mellin inversion, we get h1(x)=3ζ(2)w^(0)12πi∫(32)∑s≥1soddμ(s)s1+z(2−z−1)Mg^(z)dzxz=3ζ(2)w^(0)12πi∫(32)(2−z−1)(1−2−1−z)ζ(1+z)Mg^(z)dzxz. Similarly, h2(x)=3ζ(2)w^(0)12πi∫(−12)(2−z−1−1)(1−2−2−z)ζ(2+z)Mg(−z)xzdz, and shifting the contour of integration to the left, we obtain h2(x)=3ζ(2)w^(0)12πi∫(−54)(2−z−1−1)(1−2−2−z)ζ(2+z)Mg(−z)xzdz. Hence, we write (2.1) as J(X)=3ζ(2)Lw^(0)12πi∫0∞(ϕ^(1+τL)∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)dze(z−1)τ/2+ϕ^(1−τL)∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)ezτ/2dz)dτ=3ζ(2)Lw^(0)12πi(∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)∫0∞ϕ^(1+τL)e−(z−1)τ/2dτdz+∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)∫0∞ϕ^(1−τL)ezτ/2dτdz). (2.2) Now we use the Taylor expansions of ϕ^(1+τL) and ϕ^(1−τL) in (2.2). From the constant terms in these expansions, we obtain 3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)dzz−1−∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)=3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(12)(2−z−1−1)ζ(z+1)(1−2−2−z)ζ(2+z)Mg^(z+1)dzz−∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)=3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(12)(2−z−1−1)ζ(z+1)(1−2−2−z)ζ(2+z)Mg^(z+1)dzz−∫(12)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)+ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx)=3ζ(2)Lw^(0)2ϕ^(1)2πi∫(12)(2−z−1−1)(1−2−2−z)ζ(2+z)(ζ(z+1)Mg^(z+1)−ζ(−z)Mg(−z))dzz+ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx). Next we note that ζ(z+1)Mg^(z+1)=ζ(−z)Mg(−z). (2.3) Indeed, applying Plancherel’s identity, we have that Mg^(z+1)=∫0∞xzg^(x)dx=12∫R∣x∣zg^(x)dx=−sin(πz/2)Γ(z+1)(2π)z+1∫Rg(x)∣x∣z+1dx=−2sin(πz/2)Γ(z+1)(2π)z+1Mg(−z), where the Fourier transform of ∣x∣z is in the sense of distributions (cf. [9, Exercise 8.7]). Hence, (2.3) follows by an application of the functional equation of the zeta function. We conclude that J(X)=ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx)+O(L−2) (cf. the proof of [8, Lemma 3.6]). Now we consider the integral in the above formula. For small η>0, we have that ∫0∞(logx)g′(x)dx=∫0∞8πex(logx)w^′(4πex2)dx=12∫0∞log(x4πe)w^′(x)dx=w^(0)2log(4πe)+12∫η∞(logx)w^′(x)dx+O(ηlog(η−1))=w^(0)2log(4πe)+12[(logx)w^(x)]η∞−12∫η∞w^(x)xdx+O(ηlog(η−1))=w^(0)2log(4πe)+w^(0)2log(η−1)−12∫η∞w^(x)xdx+O(ηlog(η−1))=w^(0)2log(4πe)−12∫η∞w^(x)−w^(0)I[0,1](x)xdx+O(ηlog(η−1)). Hence, by taking the limit as η tends to zero and using [21, Example (e) on page 132], we obtain ∫0∞(logx)g′(x)dx=w^(0)2log(4πe)−14∫Rw^(x)−w^(0)I[−1,1](x)∣x∣dx=w^(0)2log(4πe)+12∫R(γ+log∣2πx∣)w(x)dx=w^(0)2log(23π2e1+γ)+∫0∞(logx)w(x)dx. We conclude that J(X)=ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)(w^(0)2log(23π2e1+γ)+∫0∞(logx)w(x)dx))+O(L−2)=ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). This verifies the identity (1.4) and thus completes the proof of Theorem 1.1. 3. The Ratios Conjecture’s prediction 3.1. The Ratios Conjecture In this section, we formulate an appropriate version of the Ratios Conjecture. Such a calculation was already performed by Conrey and Snaith [3] in the family of L-functions associated to even real Dirichlet characters. For our purposes, we need to derive an analogous conjecture with an additional smooth weight function for the family F*(X). To begin, we consider the sum R(α,γ)≔1W*(X)∑*doddw(dX)L(12+α,χ8d)L(12+γ,χ8d). (3.1) For L(s,χ8d), we have 1L(s,χ8d)=∑n=1∞μ(n)χ8d(n)ns (3.2) and the approximate functional equation L(s,χ8d)=∑n<xχ8d(n)ns+(π8∣d∣)s−12Γ(1+a−s2)Γ(a+s2)∑n<yχ8d(n)n1−s+Error, (3.3) where xy=4∣d∣/π and a≔a(d)={0ifd≥0,1ifd<0. (3.4) We now follow the Ratios Conjecture recipe [2] (see also the presentation in [3]) and disregard the error term and complete the sums (that is, replace x and y with infinity). The first step is to replace the numerator of (3.1) with the approximate functional equation (3.3) (with the above modification) and the denominator of (3.1) with (3.2). We first focus on the principal sum from (3.3) evaluated at s=12+α, which gives the contribution R1(α,γ)≔1W*(X)∑*doddw(dX)∑h,mμ(h)χ8d(hm)h12+γm12+α (3.5) to (3.1). The next step in the Ratios Conjecture procedure is to replace χ8d(hm) in (3.5) with its weighted average over the family F*(X). From [8, Lemma 2.2 and Remark 2.3], we have that 1W*(X)∑*doddw(dX)χ8d(hm)={∏p∣hm(pp+1)+Oh,m,ε(X−34+ε)ifhm=odd□,Oh,m,ε(X−34+ε)otherwise. (3.6) Thus, the main contribution to the sum in (3.5) occurs when hm is an odd square and, following the recipe, we disregard the non-square terms and the error terms. Hence, (3.5) is replaced with R˜1(α,γ)≔∑hm=odd□μ(h)h12+γm12+α∏p∣hmpp+1 and writing this as an Euler product gives R˜1(α,γ)=∏p>2(1+pp+1∑k,ℓ≥0k+ℓ>0k+ℓevenμ(pk)pk(12+γ)+ℓ(12+α))=∏p>2(1+pp+1(∑ℓ=1∞1pℓ(1+2α)−1p1+α+γ∑ℓ=0∞1pℓ(1+2α)))=∏p>2(1+p(p+1)(1−p−1−2α)(1p1+2α−1p1+α+γ)). In the above Euler product, we factor out zeta functions corresponding to the divergent parts of R˜1(α,γ) (as α,γ→0). This results in R˜1(α,γ)=ζ(1+2α)ζ(1+α+γ)A(α,γ), (3.7) where A(α,γ)≔(21+α+γ−2γ−α21+α+γ−1)∏p>2(1−1p1+α+γ)−1·(1−1(p+1)p1+2α−1(p+1)pα+γ). (3.8) Note that the product A(α,γ) is absolutely convergent for R(α),R(γ)>−14. Next we consider the contribution of the dual sum coming from the approximate functional equation (the second sum in (3.3)) to (3.1), namely the sum R2(α,γ)≔1W*(X)∑*doddw(dX)Xd(12+α)∑h,mμ(h)χ8d(hm)h12+γm12−α, (3.9) where Xd(s)≔Γ(1+a−s2)Γ(a+s2)(π8∣d∣)s−12. (3.10) As above we follow the Ratios Conjecture procedure and replace χ8d(hm) in (3.9) with its weighted average over the family F*(X). Using (3.6), we replace (3.9) with R˜2(α,γ)≔1W*(X)∑*doddw(dX)Xd(12+α)R˜1(−α,γ). Finally, using the formula (3.7), we state the Ratios Conjecture for our weighted family of quadratic Dirichlet L-functions as: Conjecture 3.1 Let ε>0and let wbe an even and nonnegative Schwartz test function on Rwhich is not identically zero. Assume GRH and suppose that the complex numbers αand γsatisfy ∣R(α)∣<14, 1logX≪R(γ)<14and I(α),I(γ)≪X1−ε. Then we have that 1W*(X)∑*doddw(dX)L(12+α,χ8d)L(12+γ,χ8d)=ζ(1+2α)ζ(1+α+γ)A(α,γ)+1W*(X)∑*doddw(dX)Xd(12+α)ζ(1−2α)ζ(1−α+γ)A(−α,γ)+Oε(X−12+ε),where A(α,γ)is defined in (3.8) and Xd(s)is defined in (3.10). (The error term Oε(X−12+ε)is part of the statement of the Ratios Conjecture. Note also that the extra conditions on αand γ, which were not used in the derivation of Conjecture3.1, are included here as standard conditions under which conjectures produced by the Ratios Conjecture recipe are expected to hold.) In our calculation of the 1-level density (see Section 3.2), we require the average of the logarithmic derivative of the L-functions in F*(X). We set Aα(r,r)≔∂∂αA(α,γ)∣α=γ=r. (3.11) Lemma 3.2 Let ε>0and let wbe an even and nonnegative Schwartz test function on Rwhich is not identically zero. Suppose that r∈Csatisfies 1logX≪R(r)<14and I(r)≪X1−ε. Then, assuming GRH and Conjecture3.1, we have that 1W*(X)∑*doddw(dX)L′(12+r,χ8d)L(12+r,χ8d)=ζ′(1+2r)ζ(1+2r)+Aα(r,r)−1W*(X)∑*doddw(dX)Xd(12+r)·ζ(1−2r)A(−r,r)+Oε(X−12+ε). (3.12) Proof Observing that A(r,r)=1, we get ∂∂αζ(1+2α)ζ(1+α+γ)A(α,γ)∣α=γ=r=ζ′(1+2r)ζ(1+2r)+Aα(r,r) and ∂∂α1W*(X)∑*doddw(dX)Xd(12+α)ζ(1−2α)ζ(1−α+γ)A(−α,γ)∣α=γ=r=−1W*(X)∑*doddw(dX)Xd(12+r)ζ(1−2r)A(−r,r), which gives the main term in (3.12). Finally, a straightforward argument using Cauchy’s integral formula for derivatives shows that the error term remains the same under differentiation. □ 3.2. The Ratios Conjecture’s prediction for the 1-level density In this section, we derive a first formulation of the Ratios Conjecture’s prediction for the 1-level density of low-lying zeros in the family F*(X). Proof of Theorem 1.3 We recall that D*(ϕ;X)=1W*(X)∑*doddw(dX)∑γ8dϕ(γ8dL2π). Using the argument principle, we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi(∫(c)−∫(1−c))L′(s,χ8d)L(s,χ8d)ϕ(−iL2π(s−12))ds (3.13) with 12+1logX<c<34. For the integral in (3.13) on the line with real part 1−c, we make the change of variables s↦1−s. Recalling that ϕ is even, we find that this integral equals 1W*(X)∑*doddw(dX)12πi∫(c)L′(1−s,χ8d)L(1−s,χ8d)ϕ(−iL2π(s−12))ds. (3.14) Next, applying the functional equation Λ(s,χ8d)≔(8∣d∣π)12(s+a)Γ(s+a2)L(s,χ8d)=Λ(1−s,χ8d) (cf. [5, Section 9]), together with (3.10), we obtain L′(s,χ8d)L(s,χ8d)=Xd′(s)Xd(s)−L′(1−s,χ8d)L(1−s,χ8d). (3.15) Using (3.14) and (3.15) together with the change of variables s=12+r, we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c−12)(2L′(12+r,χ8d)L(12+r,χ8d)−Xd′(12+r)Xd(12+r))ϕ(iLr2π)dr. (3.16) Changing the order of summation and integration in (3.16), we substitute 1W*(X)∑*doddw(dX)L′(12+r,χ8d)L(12+r,χ8d) with the right-hand side of (3.12). Note that this substitution is valid only when I(r)<X1−ε. However, since ϕ^ has compact support on R, the function ϕ(iLr2π) is rapidly decaying as ∣I(r)∣→∞. From this fact and the estimate [11, Theorem 5.17] of the logarithmic derivative of Dirichlet L-functions, we can bound the tail of the integral in (3.16) by Oε(X−1+ε). Furthermore, using a similar argument to bound the tail of the integral in (3.17), we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c−12)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)−Xd′(12+r)Xd(12+r)−2Xd(12+r)ζ(1−2r)A(−r,r))ϕ(iLr2π)dr+Oε(X−12+ε). (3.17) The final step is to move the contour of integration from R(r)=c−12=c′ to R(r)=0. Note that the function 2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)−Xd′(12+r)Xd(12+r)−2Xd(12+r)ζ(1−2r)A(−r,r) is analytic in this region. Thus, by Cauchy’s Theorem, we have that D*(ϕ;X)=1W*(X)∑*doddw(dX)12π∫R(2ζ′(1+2it)ζ(1+2it)+2Aα(it,it)+log(8∣d∣π)+12Γ′Γ(14+a−it2)+12Γ′Γ(14+a+it2)−2Xd(12+it)ζ(1−2it)A(−it,it))ϕ(tL2π)dt+Oε(X−12+ε), since Xd′(12+it)Xd(12+it)=log(π8∣d∣)−12Γ′Γ(14+a−it2)−12Γ′Γ(14+a+it2). This completes the proof.□ 4. Making the Ratios Conjecture’s prediction explicit In this section, we wish to compare the Ratios Conjecture’s prediction for D*(ϕ;X) in Theorem 1.3 with the result in Theorem 1.1. We recall the prediction obtained in (3.17), which for 1logX<c′<14 equals D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)+log(8∣d∣π)+12Γ′Γ(14+a−r2)+12Γ′Γ(14+a+r2)−2Xd(12+r)ζ(1−2r)A(−r,r))ϕ(iLr2π)dr+Oε(X−12+ε). (4.1) To begin, we focus on the first two terms in (4.1). Recalling (3.8), we write A(α,γ)=(1+21+α−21+γ3·21+2α+γ−2γ−21+α)∏p(1−1p1+α+γ)−1·(1−1(p+1)p1+2α−1(p+1)pα+γ). Using the fact that A(r,r)=1, we compute Aα(r,r)=2log23(21+2r−1)+∑plogp(p+1)(p1+2r−1). Furthermore, we have that ζ′(1+2r)ζ(1+2r)=−∑plogpp1+2r−1. We now have the following result. Lemma 4.1 Let ε>0and suppose that 1logX<c′<14. Then 12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r))ϕ(iLr2π)dr=−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL). (4.2) Remark 4.2 In [8, Lemma 3.7], it was shown that the right-hand side of (4.2) is asymptotic to −ϕ(0)/2 as X→∞. The Katz–Sarnak prediction (1.2) is obtained by combining this term with the main terms occurring in Lemmas 4.3 and 4.6. Proof of Lemma 4.1 We have that 12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r))ϕ(iLr2π)dr=1πi∫(c′)(−∑plogpp1+2r−1+2log23(21+2r−1)+∑plogp(p+1)(p1+2r−1))ϕ(iLr2π)dr=1πi∫(c′)(2log23∑j=1∞12(1+2r)j−∑plogp∑j=1∞1p(1+2r)j+∑plogp(p+1)∑j=1∞1p(1+2r)j)ϕ(iLr2π)dr=1πi∫(c′)(2log23∑j=1∞12(1+2r)j−∑pplogpp+1∑j=1∞1p(1+2r)j)ϕ(iLr2π)dr. (4.3) Making the substitution u=−iLr2π, we have that (4.3) becomes 2L∫C′(2log23∑j=1∞e−(log2)(4πiujL)2j−∑pplogpp+1∑j=1∞e−(logp)(4πiujL)pj)ϕ(u)du, (4.4) where C′ denotes the horizontal line I(u)=−Lc′2π. We note that the summations inside the integral over C′ in (4.4) converge absolutely and uniformly on compact subsets. Thus, we may interchange the order of integration and summation and (4.4) becomes 4log23L∑j=1∞12j∫C′ϕ(u)e−2πiu(2jlog2L)du−2L∑pplogpp+1∑j=1∞1pj∫C′ϕ(u)e−2πiu(2jlogpL)du. Next, we move the contour of integration from C′ to the line I(u)=0. Note that this is allowed since ϕ^ has compact support on R and the entire function ϕ(z)≔∫Rϕ^(x)e2πixzdx satisfies the inequality ∣ϕ(T+it)∣≤12π∣T∣∫R∣ϕ^′(x)∣max(1,exLc′)dx, uniformly for −Lc′2π≤t≤0, as T→±∞. Thus, (4.3) equals −2L∑p>2plogpp+1∑j=1∞1pjϕ^(2jlogpL), which concludes the proof.□ We now study the third, fourth and fifth terms in (4.1). For these terms, we can shift the line of integration to the imaginary axis, since the integrands are analytic in this region. We now give estimates for these shifted integrals. Lemma 4.3 Fix ε>0. We have 1W*(X)∑*doddw(dX)12π∫Rlog(8∣d∣π)ϕ(tL2π)dt=ϕ^(0)+ϕ^(0)L(log(24e)+2w^(0)∫0∞w(x)(logx)dx)+Oε,w(X−12+ε). Proof The result follows as in [8, Lemma 2.4] (see also [7, Lemma 2.8]).□ Lemma 4.4 We have 1W*(X)∑*doddw(dX)14π∫R(Γ′Γ(14+a−it2)+Γ′Γ(14+a+it2))ϕ(tL2π)dt=ϕ^(0)Llog(2−3e−γ)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx. Proof We have that 1W*(X)∑*doddw(dX)14π∫R(Γ′Γ(14+a−it2)+Γ′Γ(14+a+it2))ϕ(tL2π)dt=14L∫R(Γ′Γ(14+πiuL)+Γ′Γ(14−πiuL)+Γ′Γ(34+πiuL)+Γ′Γ(34−πiuL))ϕ(u)du=(Γ′Γ(14)+Γ′Γ(34))ϕ^(0)2L+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx, from [16, Lemma 12.14]. The result follows.□ The next lemma is required to evaluate the last term in (4.1). Lemma 4.5 Let ε>0and assume that 0≤R(r)≤12. Then we have the estimate 1W*(X)∑*doddw(dX)∣d∣−r=2w^(0)X−rMw(1−r)+Oε,w((∣I(r)∣+1)1384+εX−12−R(r)+ε). Proof We write the sum we are interested in as the Mellin integral S≔∑*doddw(dX)∣d∣−r=12πi∫(2)2s+r+12s+r+1ζ(s+r)ζ(2(s+r))XsMw(s)ds. We pull the contour of integration to the line R(s)=12−R(r)+ε. Then we have a contribution from the simple pole at s=1−r. Note that the restriction on R(r) ensures that we do not encounter the potential pole of Mw(s) at s=0. Hence, by the rapid decay of Mw on vertical lines, Bourgain’s subconvexity bound [1, Theorem 5] on ∣ζ(s+r)∣ and the boundedness of ∣ζ(2(s+r))−1∣, we have that S=43ζ(2)X1−rMw(1−r)+Oε,w((∣I(r)∣+1)1384+εX12−R(r)+ε). Similarly as in [7, Lemma 2.10], it can be shown that W*(X)=23ζ(2)Xw^(0)+Oε,w(X12+ε). We complete the proof by combining the last two estimates.□ We now give an asymptotic formula for the last term in (4.1). By a straightforward computation and by recalling the definitions of Xd and A, we apply the substitution r=2πiτ/L to obtain I≔−ζ(2)L∫C′(Γ(14−πiτL)Γ(14+πiτL)+Γ(34−πiτL)Γ(34+πiτL))(π8)2πiτL(1+2−24πiτL+14−24πiτL)ζ(1−4πiτL)ζ(2−4πiτL)ϕ(τ)×1W*(X)∑*doddw(dX)∣d∣−2πiτLdτ, (4.5) where C′ again denotes the horizontal line I(τ)=−Lc′2π. Lemma 4.6 We have the asymptotic formula I=ϕ(0)2−12∫−11ϕ^(τ)dτ+ϕ^(1)L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). Proof Let η>0 be small. We first change the contour of integration in I to the path C≔C0∪C1∪C2, where C0≔{I(τ)=0,∣R(τ)∣≥Lε},C1≔{I(τ)=0,η≤∣R(τ)∣≤Lε},C2≔{∣τ∣=η,I(τ)≤0}. For the part of the integral I over C0, we trivially bound the sum over d and use the rapid decay of ϕ to obtain the bound ≪ε1L∫C0max(log(∣τ∣+2),L∣τ∣−1)(∣τ∣+2)−3/ε2dτ≪∫LεLlog(∣τ∣+2)∣τ∣−1(∣τ∣+2)−3/ε2dτ+L−2/ε2≪L−3/ε(logL)2+L−2/ε2≪L−2/ε. On C1∪C2, we use Taylor expansions for each factor in the integrand of I, except the last in which we apply Lemma 4.5. This yields −ζ(2)L(Γ(14−πiτL)Γ(14+πiτL)+Γ(34−πiτL)Γ(34+πiτL))(π8)2πiτL×(1+2−24πiτL+14−24πiτL)ζ(1−4πiτL)ζ(2−4πiτL)ϕ(τ)W*(X)∑*doddw(dX)∣d∣−2πiτL=−1L(2+(γ+log8)4πiτL+O(∣τ∣2L2))(1+2πiτLlog(π8)+O(∣τ∣2L2))×(1+(2ζ′(2)ζ(2)−43log2)2πiτL+O(∣τ∣2L2))(−L4πiτ+γ+O(∣τ∣L))ϕ(τ)×2w^(0)(X−2πiτLMw(1−2πiτL)+Oε,w((∣I(iτ)∣L+1)1384+εX−12+ε))=12πiτ(1+2πiτL(−γ−log(243π)+2ζ′(2)ζ(2))+O(∣τ∣2L2))ϕ(τ)e−2πiτ×(1−2πiτlog(2πe)L+O(∣τ∣2L2))(1−Mw′(1)Mw(1)2πiτL+O(∣τ∣2L2))+Oε,w(X−12+ε)=12πiτ(1+2πiτL(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(∣τ∣2L2))ϕ(τ)e−2πiτ+Oε,w(X−12+ε). Hence I=I1+I2+Ow(L−2), where I1≔∫C1∪C212πiτϕ(τ)e−2πiτdτ and I2≔1L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))∫C1∪C2ϕ(τ)e−2πiτdτ. For the second integral I2, by the rapid decay of ϕ on the real line and the holomorphy of the integrand, we have that I2=1L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))∫Rϕ(τ)e−2πiτdτ+Oε(L−2/ε)=ϕ^(1)L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+Oε(L−2/ε). Similarly, for the first integral I1, we find that I1=12πi∫C0∪C1∪C2e−2πiτϕ(τ)τdτ+Oε(L−2/ε)=J1+J2+Oε(L−2/ε), where J1≔12πi∫C0∪C1∪C2cos(2πτ)ϕ(τ)τdτ and J2≔−12π∫C0∪C1∪C2sin(2πτ)ϕ(τ)τdτ. We see that since the integrand in J1 is odd, the part of the integral on C0∪C1 is zero. Hence, J1=12πi∫C2cos(2πτ)ϕ(τ)τdτ, which by the residue theorem tends to ϕ(0)/2 as η tends to zero. As for the integral J2, we apply Plancherel’s identity. Since sin(2πτ)/τ is an entire function, J2=−∫Rsin(2πτ)2πτϕ(τ)dτ=−12∫RI[−1,1](τ)ϕ^(τ)dτ=−12∫−11ϕ^(τ)dτ, which coincides with the second term in the Katz–Sarnak prediction. Since all of our error terms are independent of η, we conclude the desired result.□ Proof of Theorem 1.4 The proof follows by combining Lemmas 4.1, 4.3, 4.4 and 4.6.□ Funding The first author was supported by an NSERC discovery grant. The third author was supported by a grant from the Swedish Research Council (Grant 2016-03759). Acknowledgements We thank Bruno Martin for helpful discussions. We also thank the referee for pointing our attention to the papers [13, 14] of Mason and Snaith. References 1 J. Bourgain , Decoupling, exponential sums and the Riemann zeta function , J. Amer. Math. Soc. 30 ( 2017 ), 205 – 224 . Google Scholar CrossRef Search ADS 2 J. B. Conrey , D. W. Farmer and M. R. Zirnbauer , Autocorrelation of ratios of L-functions , Commun. Number Theory Phys. 2 ( 2008 ), 593 – 636 . Google Scholar CrossRef Search ADS 3 J. B. Conrey and N. C. Snaith , Applications of the L-functions ratios conjectures , Proc. Lond. Math. Soc. (3) 94 ( 2007 ), 594 – 646 . Google Scholar CrossRef Search ADS 4 J. B. Conrey and N. C. Snaith , Correlations of eigenvalues and Riemann zeros , Commun. Number Theory Phys. 2 ( 2008 ), 477 – 536 . Google Scholar CrossRef Search ADS 5 H. Davenport , Multiplicative number theory, third edition, revised and with a preface by H. L. Montgomery, Graduate Texts in Mathematics Vol. 74, Springer-Verlag , New York , 2000 . 6 A. Entin , E. Roditty-Gershon and Z. Rudnick , Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and random matrix theory , Geom. Funct. Anal. 23 ( 2013 ), 1230 – 1261 . Google Scholar CrossRef Search ADS 7 D. Fiorilli , J. Parks and A. Södergren , Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture , Math. Proc. Cambridge Philos. Soc. 160 ( 2016 ), 315 – 351 . Google Scholar CrossRef Search ADS 8 D. Fiorilli , J. Parks and A. Södergren , Low-lying zeros of quadratic Dirichlet L-functions: Lower order terms for extended support , Compos. Math. 153 ( 2017 ), 1196 – 1216 . Google Scholar CrossRef Search ADS 9 F. G. Friedlander , Introduction to the theory of distributions, second edition, with additional material by M. Joshi , Cambridge University Press , Cambridge , 1998 . 10 P. Gao , n-level density of the low-lying zeros of quadratic Dirichlet L-functions, Ph.D. Thesis, University of Michigan, 2005 . 11 H. Iwaniec and E. Kowalski , Analytic number theory, American Mathematical Society Colloquium Publications Vol. 53 , American Mathematical Society , Providence, RI , 2004 . 12 N. M. Katz and P. Sarnak , Zeroes of zeta functions and symmetry , Bull. Amer. Math. Soc. (N.S.) 36 ( 1999 ), 1 – 26 . Google Scholar CrossRef Search ADS 13 A. M. Mason and N. C. Snaith , Symplectic n-level densities with restricted support , Random Matrices Theory Appl. 5 ( 2016 ), 1650013 , 36 pp. Google Scholar CrossRef Search ADS 14 A. M. Mason and N. C. Snaith , Orthogonal and symplectic n-level densities , Mem. Amer. Math. Soc. 251 ( 2018 ), 93 pp . 15 S. J. Miller , A symplectic test of the L-functions ratios conjecture , Int. Math. Res. Not. IMRN2008 ( 2008 ), 36 pp . Art. IDrnm146. 16 H. L. Montgomery and R. C. Vaughan , Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics Vol. 97, Cambridge University Press , Cambridge , 2007 . 17 A. E. Özlük and C. Snyder , On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis , Acta Arith. 91 ( 1999 ), 209 – 228 . Google Scholar CrossRef Search ADS 18 M. Rubinstein , Low-lying zeros of L-functions and random matrix theory , Duke Math. J. 109 ( 2001 ), 147 – 181 . Google Scholar CrossRef Search ADS 19 Z. Rudnick , Traces of high powers of the Frobenius class in the hyperelliptic ensemble , Acta Arith. 143 ( 2010 ), 81 – 99 . Google Scholar CrossRef Search ADS 20 P. Sarnak , S. W. Shin and N. Templier , Families of L-functions and their symmetry, Proceedings of Simons Symposia, Families of Automorphic Forms and the Trace Formula, Springer-Verlag ( 2016 ), 531–578. 21 V. S. Vladimirov , Equations of mathematical physics, translated from the Russian by Audrey Littlewood, Pure and Applied Mathematics Vol. 3, Marcel Dekker, Inc , New York , 1971 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. 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Low-lying zeros of quadratic Dirichlet L-functions: A transition in the ratios conjecture

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Abstract

Abstract We study the 1-level density of low-lying zeros of quadratic Dirichlet L-functions by applying the L-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz–Sarnak heuristic as well as in the lower-order terms when the support of the Fourier transform of the corresponding test function reaches the point 1. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case. 1. Introduction In this paper, we study low-lying zeros in the family F*(X)≔{L(s,χ8d):1≤∣d∣≤X;disoddandsquarefree} of Dirichlet L-functions. Here we define the real primitive character of conductor 8∣d∣ by the Kronecker symbol χ8d(n)≔(8dn). Low-lying zeros for quadratic Dirichlet L-functions have been studied extensively in the literature, see, for example, Özlük and Snyder [17], Rubinstein [18], Gao [10], Entin et al. [6], Miller [15] and the authors’ previous paper [8]. The L-functions Ratios Conjecture of Conrey et al. [2, Section 5] is a recipe for obtaining conjectural formulas for averages of quotients of (products of) L-functions evaluated at certain values in the critical strip. This can in turn be used to give extremely precise predictions for a variety of statistics for families of L-functions. For example, this has been carried out notably by Conrey and Snaith [3, 4] for correlations of the zeros of the Riemann zeta function and the 1-level density and moments of L-functions, and more recently by Mason and Snaith [14] for the n-level density of orthogonal and symplectic families of L-functions. Our goal is to compare the Ratios Conjecture prediction of [3] with the estimate we obtained in [8] for the 1-level density in the family F*(X) of quadratic Dirichlet L-functions. While agreement between the leading term in the Ratios Conjecture and the Katz–Sarnak random matrix theory prediction [12] was shown previously in [13, 15], the novelty in our work is the isolation of a sharp transition in the lower-order terms that we give explicitly and which agrees with Rudnick’s work [19] over function fields and our previous results [8]. Before we describe the new results, we first review and refine our previous work. We begin by introducing the 1-level density of the family F*(X). Given a large positive number X, we set L≔log(X2πe) and W*(X)≔∑*doddw(dX), where w(t) is an even, non-zero and nonnegative Schwartz function and the star on the sum denotes a restriction to squarefree integers. We introduce the 1-level density of the family F*(X) as the functional D*(ϕ;X)≔1W*(X)∑*doddw(dX)∑γ8dϕ(γ8dL2π). (1.1) Here and throughout, ϕ will be a real and even Schwartz test function. Furthermore, we define γ8d≔−i(ρ8d−12), where ρ8d runs over the non-trivial zeros of L(s,χ8d) (that is, zeros with 0<R(ρ8d)<1). The Katz–Sarnak heuristic [12] provides a precise prediction of the statistics of low-lying zeros in families of L-functions (see also the recent paper [20]). In our situation, the heuristic asserts that F*(X) has symplectic symmetry type and in particular that limX→∞D*(ϕ;X)=ϕ^(0)−12∫−11ϕ^(u)du (1.2) independently of the support of ϕ^. Moreover, note that there is a phase transition in the right-hand side of (1.2) occurring when the supremum σ of the support of ϕ^ reaches 1. This transition has a strong influence on the shape of the lower order terms in D*(ϕ;X) (cf. [8, 19]) and will thus be of fundamental interest in the current paper. The lower order terms in the 1-level density for the family F*(X) were recently studied under the assumption of GRH in [8]. There we obtained an asymptotic formula for D*(ϕ;X) in descending powers of logX, which is valid when the support of ϕ^ is contained in (−2,2). In particular, we uncovered a phase transition when σ approaches 1 in the main term as well as in the lower order terms. This asymptotic formula (see [8, Theorem 3.5]) contains a term J(X) which is expressed in terms of explicit transforms of the weight function w. In the current paper, we give an asymptotic for this complicated expression (see Section 2), and, as a consequence, the following result holds. Theorem 1.1 Fix ε>0. Assume GRH and suppose that σ=sup(suppϕ^)<2. Then the 1-level density of low-lying zeros in the family F*(X)of quadratic Dirichlet L-functions whose conductor is an odd squarefree multiple of 8 is given by D*(ϕ;X)=ϕ^(0)+∫1∞ϕ^(u)du+ϕ^(0)L(log(2e1−γ)+2w^(0)∫0∞w(x)(logx)dx)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL)+J(X)+Oε(Xσ6−13+ε), (1.3)where J(X)is defined in (2.1) and satisfies the asymptotic relation J(X)=ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). (1.4) (Here Mwdenotes the Mellin transform of w.) Remark 1.2 Note that Theorem 1.1 is in agreement with the Katz–Sarnak heuristic (1.2) (cf. [8, Section 3]). In fact, in [8, Theorem 1.1], we show how to write (1.3) as D*(ϕ;X)=ϕ^(0)−12∫−11ϕ^(u)du+∑k=1KRw,k(ϕ)(logX)k+Ow,ϕ,K(1(logX)K+1), where Rw,k(ϕ) are linear functionals in ϕ that can be given explicitly in terms of w and the derivatives of ϕ^ at the points 0 and 1. An analogous expression having a transition at the point 1 was obtained by Rudnick [19, Corollary 3] in the function field case. In Conjecture 3.1, we adapt the Ratios Conjecture [3, Conjecture 2.6] for the family F*(X) to our situation (we need to add and keep track of the smooth weight function w). Following [3, Theorem 3.1], we use this conjecture to obtain the following asymptotic expression for D*(ϕ;X) (see Section 3). Theorem 1.3 Fix ε>0and let ϕbe an even Schwartz test function on Rwhose Fourier transform has compact support. Assume GRH and Conjecture3.1 (the Ratios Conjecture for F*(X)). Then the 1-level density for the low-lying zeros in the family F*(X)is given by D*(ϕ;X)=1W*(X)∑*doddw(dX)12π∫R(2ζ′(1+2it)ζ(1+2it)+2Aα(it,it)+log(8∣d∣π)+12Γ′Γ(14+a−it2)+12Γ′Γ(14+a+it2)−2Xd(12+it)ζ(1−2it)A(−it,it))·ϕ(tL2π)dt+Oε(X−12+ε),where A, Aα, aand Xdare defined by (3.8), (3.11), (3.4) and (3.10), respectively. Given the apparent difference between the formulas for D*(ϕ;X) in Theorems 1.1 and 1.3, it is an interesting question to ask whether the detailed information about the phase transition at σ=1 occurring in (1.3) is also present in Theorem 1.3. Our main theorem answers this question in the affirmative and, to the best of our knowledge, this is the first comprehensive investigation of such a transition via a Ratios Conjecture calculation (see Section 4). Theorem 1.4 Let ϕbe an even Schwartz test function on Rwhose Fourier transform has compact support. Assume GRH and Conjecture3.1 (the Ratios Conjecture for F*(X)). Then the 1-level density for the low-lying zeros in the family F*(X)can be written in the form D*(ϕ;X)=ϕ^(0)+∫1∞ϕ^(u)du+ϕ^(0)L(log(2e1−γ)+2w^(0)∫0∞w(x)(logx)dx)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL)+ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2).In particular, for test functions ϕwith σ=sup(suppϕ^)<2, this formula agrees with Theorem1.1up to an error term of order O(L−2). Remark 1.5 The error term in Theorem 1.4 results from a Taylor expansion of the integrand in (4.5) in powers of τL with remainder O(∣τ∣2L−2). In principle, with a higher order Taylor expansion, one can obtain an error term in Theorem 1.4 of size O(L−k) for any k≥2 by the same methods. However, for conciseness, we choose not to pursue this here. Remark 1.6 The first indications of a phase transition in a Ratios Conjecture calculation were given by Miller [15, Section 2.2]. However, the arguments in [15] contain several serious issues. In particular, the Katz–Sarnak main term was not computed correctly (cf. [15, Lemma 2.1, Lemma 2.6]). From the results in [8, 19], it is not clear whether or not we can expect the first lower order term to have the same shape also for test functions of larger support of their Fourier transforms. One of the consequences of Theorem 1.4 is that under the assumption of the Ratios Conjecture no additional phase transitions occur when the support of the Fourier transform reaches points in the interval [2,∞). 2. Proof of Theorem 1.1 As in [8] we define J(X)≔1L∫0∞(ϕ^(1+τL)eτ2∑n≥1h1(neτ2)+ϕ^(1−τL)∑n≥1h2(neτ2))dτ, (2.1) where h1(x)≔3ζ(2)w^(0)∑s≥1soddμ(s)s(g^(2sx)−g^(sx)),h2(x)≔3ζ(2)w^(0)∑s≥1soddμ(s)s2(12g(x2s)−g(xs)), and g(y)≔w^(4πey2). The formula (1.3) for D*(ϕ;X) was obtained in [8, Theorem 3.5]. Thus, it remains to prove the non-trivial estimate (1.4) for J(X). Using Mellin inversion, we get h1(x)=3ζ(2)w^(0)12πi∫(32)∑s≥1soddμ(s)s1+z(2−z−1)Mg^(z)dzxz=3ζ(2)w^(0)12πi∫(32)(2−z−1)(1−2−1−z)ζ(1+z)Mg^(z)dzxz. Similarly, h2(x)=3ζ(2)w^(0)12πi∫(−12)(2−z−1−1)(1−2−2−z)ζ(2+z)Mg(−z)xzdz, and shifting the contour of integration to the left, we obtain h2(x)=3ζ(2)w^(0)12πi∫(−54)(2−z−1−1)(1−2−2−z)ζ(2+z)Mg(−z)xzdz. Hence, we write (2.1) as J(X)=3ζ(2)Lw^(0)12πi∫0∞(ϕ^(1+τL)∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)dze(z−1)τ/2+ϕ^(1−τL)∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)ezτ/2dz)dτ=3ζ(2)Lw^(0)12πi(∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)∫0∞ϕ^(1+τL)e−(z−1)τ/2dτdz+∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)∫0∞ϕ^(1−τL)ezτ/2dτdz). (2.2) Now we use the Taylor expansions of ϕ^(1+τL) and ϕ^(1−τL) in (2.2). From the constant terms in these expansions, we obtain 3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(32)(2−z−1)ζ(z)(1−2−1−z)ζ(1+z)Mg^(z)dzz−1−∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)=3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(12)(2−z−1−1)ζ(z+1)(1−2−2−z)ζ(2+z)Mg^(z+1)dzz−∫(−54)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)=3ζ(2)Lw^(0)2ϕ^(1)2πi(∫(12)(2−z−1−1)ζ(z+1)(1−2−2−z)ζ(2+z)Mg^(z+1)dzz−∫(12)(2−z−1−1)ζ(−z)(1−2−2−z)ζ(2+z)Mg(−z)dzz)+ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx)=3ζ(2)Lw^(0)2ϕ^(1)2πi∫(12)(2−z−1−1)(1−2−2−z)ζ(2+z)(ζ(z+1)Mg^(z+1)−ζ(−z)Mg(−z))dzz+ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx). Next we note that ζ(z+1)Mg^(z+1)=ζ(−z)Mg(−z). (2.3) Indeed, applying Plancherel’s identity, we have that Mg^(z+1)=∫0∞xzg^(x)dx=12∫R∣x∣zg^(x)dx=−sin(πz/2)Γ(z+1)(2π)z+1∫Rg(x)∣x∣z+1dx=−2sin(πz/2)Γ(z+1)(2π)z+1Mg(−z), where the Fourier transform of ∣x∣z is in the sense of distributions (cf. [9, Exercise 8.7]). Hence, (2.3) follows by an application of the functional equation of the zeta function. We conclude that J(X)=ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)∫0∞(logx)g′(x)dx)+O(L−2) (cf. the proof of [8, Lemma 3.6]). Now we consider the integral in the above formula. For small η>0, we have that ∫0∞(logx)g′(x)dx=∫0∞8πex(logx)w^′(4πex2)dx=12∫0∞log(x4πe)w^′(x)dx=w^(0)2log(4πe)+12∫η∞(logx)w^′(x)dx+O(ηlog(η−1))=w^(0)2log(4πe)+12[(logx)w^(x)]η∞−12∫η∞w^(x)xdx+O(ηlog(η−1))=w^(0)2log(4πe)+w^(0)2log(η−1)−12∫η∞w^(x)xdx+O(ηlog(η−1))=w^(0)2log(4πe)−12∫η∞w^(x)−w^(0)I[0,1](x)xdx+O(ηlog(η−1)). Hence, by taking the limit as η tends to zero and using [21, Example (e) on page 132], we obtain ∫0∞(logx)g′(x)dx=w^(0)2log(4πe)−14∫Rw^(x)−w^(0)I[−1,1](x)∣x∣dx=w^(0)2log(4πe)+12∫R(γ+log∣2πx∣)w(x)dx=w^(0)2log(23π2e1+γ)+∫0∞(logx)w(x)dx. We conclude that J(X)=ϕ^(1)L(log(223π2)+2ζ′(2)ζ(2)−2w^(0)(w^(0)2log(23π2e1+γ)+∫0∞(logx)w(x)dx))+O(L−2)=ϕ^(1)L(−log(273e1+γ)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). This verifies the identity (1.4) and thus completes the proof of Theorem 1.1. 3. The Ratios Conjecture’s prediction 3.1. The Ratios Conjecture In this section, we formulate an appropriate version of the Ratios Conjecture. Such a calculation was already performed by Conrey and Snaith [3] in the family of L-functions associated to even real Dirichlet characters. For our purposes, we need to derive an analogous conjecture with an additional smooth weight function for the family F*(X). To begin, we consider the sum R(α,γ)≔1W*(X)∑*doddw(dX)L(12+α,χ8d)L(12+γ,χ8d). (3.1) For L(s,χ8d), we have 1L(s,χ8d)=∑n=1∞μ(n)χ8d(n)ns (3.2) and the approximate functional equation L(s,χ8d)=∑n<xχ8d(n)ns+(π8∣d∣)s−12Γ(1+a−s2)Γ(a+s2)∑n<yχ8d(n)n1−s+Error, (3.3) where xy=4∣d∣/π and a≔a(d)={0ifd≥0,1ifd<0. (3.4) We now follow the Ratios Conjecture recipe [2] (see also the presentation in [3]) and disregard the error term and complete the sums (that is, replace x and y with infinity). The first step is to replace the numerator of (3.1) with the approximate functional equation (3.3) (with the above modification) and the denominator of (3.1) with (3.2). We first focus on the principal sum from (3.3) evaluated at s=12+α, which gives the contribution R1(α,γ)≔1W*(X)∑*doddw(dX)∑h,mμ(h)χ8d(hm)h12+γm12+α (3.5) to (3.1). The next step in the Ratios Conjecture procedure is to replace χ8d(hm) in (3.5) with its weighted average over the family F*(X). From [8, Lemma 2.2 and Remark 2.3], we have that 1W*(X)∑*doddw(dX)χ8d(hm)={∏p∣hm(pp+1)+Oh,m,ε(X−34+ε)ifhm=odd□,Oh,m,ε(X−34+ε)otherwise. (3.6) Thus, the main contribution to the sum in (3.5) occurs when hm is an odd square and, following the recipe, we disregard the non-square terms and the error terms. Hence, (3.5) is replaced with R˜1(α,γ)≔∑hm=odd□μ(h)h12+γm12+α∏p∣hmpp+1 and writing this as an Euler product gives R˜1(α,γ)=∏p>2(1+pp+1∑k,ℓ≥0k+ℓ>0k+ℓevenμ(pk)pk(12+γ)+ℓ(12+α))=∏p>2(1+pp+1(∑ℓ=1∞1pℓ(1+2α)−1p1+α+γ∑ℓ=0∞1pℓ(1+2α)))=∏p>2(1+p(p+1)(1−p−1−2α)(1p1+2α−1p1+α+γ)). In the above Euler product, we factor out zeta functions corresponding to the divergent parts of R˜1(α,γ) (as α,γ→0). This results in R˜1(α,γ)=ζ(1+2α)ζ(1+α+γ)A(α,γ), (3.7) where A(α,γ)≔(21+α+γ−2γ−α21+α+γ−1)∏p>2(1−1p1+α+γ)−1·(1−1(p+1)p1+2α−1(p+1)pα+γ). (3.8) Note that the product A(α,γ) is absolutely convergent for R(α),R(γ)>−14. Next we consider the contribution of the dual sum coming from the approximate functional equation (the second sum in (3.3)) to (3.1), namely the sum R2(α,γ)≔1W*(X)∑*doddw(dX)Xd(12+α)∑h,mμ(h)χ8d(hm)h12+γm12−α, (3.9) where Xd(s)≔Γ(1+a−s2)Γ(a+s2)(π8∣d∣)s−12. (3.10) As above we follow the Ratios Conjecture procedure and replace χ8d(hm) in (3.9) with its weighted average over the family F*(X). Using (3.6), we replace (3.9) with R˜2(α,γ)≔1W*(X)∑*doddw(dX)Xd(12+α)R˜1(−α,γ). Finally, using the formula (3.7), we state the Ratios Conjecture for our weighted family of quadratic Dirichlet L-functions as: Conjecture 3.1 Let ε>0and let wbe an even and nonnegative Schwartz test function on Rwhich is not identically zero. Assume GRH and suppose that the complex numbers αand γsatisfy ∣R(α)∣<14, 1logX≪R(γ)<14and I(α),I(γ)≪X1−ε. Then we have that 1W*(X)∑*doddw(dX)L(12+α,χ8d)L(12+γ,χ8d)=ζ(1+2α)ζ(1+α+γ)A(α,γ)+1W*(X)∑*doddw(dX)Xd(12+α)ζ(1−2α)ζ(1−α+γ)A(−α,γ)+Oε(X−12+ε),where A(α,γ)is defined in (3.8) and Xd(s)is defined in (3.10). (The error term Oε(X−12+ε)is part of the statement of the Ratios Conjecture. Note also that the extra conditions on αand γ, which were not used in the derivation of Conjecture3.1, are included here as standard conditions under which conjectures produced by the Ratios Conjecture recipe are expected to hold.) In our calculation of the 1-level density (see Section 3.2), we require the average of the logarithmic derivative of the L-functions in F*(X). We set Aα(r,r)≔∂∂αA(α,γ)∣α=γ=r. (3.11) Lemma 3.2 Let ε>0and let wbe an even and nonnegative Schwartz test function on Rwhich is not identically zero. Suppose that r∈Csatisfies 1logX≪R(r)<14and I(r)≪X1−ε. Then, assuming GRH and Conjecture3.1, we have that 1W*(X)∑*doddw(dX)L′(12+r,χ8d)L(12+r,χ8d)=ζ′(1+2r)ζ(1+2r)+Aα(r,r)−1W*(X)∑*doddw(dX)Xd(12+r)·ζ(1−2r)A(−r,r)+Oε(X−12+ε). (3.12) Proof Observing that A(r,r)=1, we get ∂∂αζ(1+2α)ζ(1+α+γ)A(α,γ)∣α=γ=r=ζ′(1+2r)ζ(1+2r)+Aα(r,r) and ∂∂α1W*(X)∑*doddw(dX)Xd(12+α)ζ(1−2α)ζ(1−α+γ)A(−α,γ)∣α=γ=r=−1W*(X)∑*doddw(dX)Xd(12+r)ζ(1−2r)A(−r,r), which gives the main term in (3.12). Finally, a straightforward argument using Cauchy’s integral formula for derivatives shows that the error term remains the same under differentiation. □ 3.2. The Ratios Conjecture’s prediction for the 1-level density In this section, we derive a first formulation of the Ratios Conjecture’s prediction for the 1-level density of low-lying zeros in the family F*(X). Proof of Theorem 1.3 We recall that D*(ϕ;X)=1W*(X)∑*doddw(dX)∑γ8dϕ(γ8dL2π). Using the argument principle, we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi(∫(c)−∫(1−c))L′(s,χ8d)L(s,χ8d)ϕ(−iL2π(s−12))ds (3.13) with 12+1logX<c<34. For the integral in (3.13) on the line with real part 1−c, we make the change of variables s↦1−s. Recalling that ϕ is even, we find that this integral equals 1W*(X)∑*doddw(dX)12πi∫(c)L′(1−s,χ8d)L(1−s,χ8d)ϕ(−iL2π(s−12))ds. (3.14) Next, applying the functional equation Λ(s,χ8d)≔(8∣d∣π)12(s+a)Γ(s+a2)L(s,χ8d)=Λ(1−s,χ8d) (cf. [5, Section 9]), together with (3.10), we obtain L′(s,χ8d)L(s,χ8d)=Xd′(s)Xd(s)−L′(1−s,χ8d)L(1−s,χ8d). (3.15) Using (3.14) and (3.15) together with the change of variables s=12+r, we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c−12)(2L′(12+r,χ8d)L(12+r,χ8d)−Xd′(12+r)Xd(12+r))ϕ(iLr2π)dr. (3.16) Changing the order of summation and integration in (3.16), we substitute 1W*(X)∑*doddw(dX)L′(12+r,χ8d)L(12+r,χ8d) with the right-hand side of (3.12). Note that this substitution is valid only when I(r)<X1−ε. However, since ϕ^ has compact support on R, the function ϕ(iLr2π) is rapidly decaying as ∣I(r)∣→∞. From this fact and the estimate [11, Theorem 5.17] of the logarithmic derivative of Dirichlet L-functions, we can bound the tail of the integral in (3.16) by Oε(X−1+ε). Furthermore, using a similar argument to bound the tail of the integral in (3.17), we obtain D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c−12)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)−Xd′(12+r)Xd(12+r)−2Xd(12+r)ζ(1−2r)A(−r,r))ϕ(iLr2π)dr+Oε(X−12+ε). (3.17) The final step is to move the contour of integration from R(r)=c−12=c′ to R(r)=0. Note that the function 2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)−Xd′(12+r)Xd(12+r)−2Xd(12+r)ζ(1−2r)A(−r,r) is analytic in this region. Thus, by Cauchy’s Theorem, we have that D*(ϕ;X)=1W*(X)∑*doddw(dX)12π∫R(2ζ′(1+2it)ζ(1+2it)+2Aα(it,it)+log(8∣d∣π)+12Γ′Γ(14+a−it2)+12Γ′Γ(14+a+it2)−2Xd(12+it)ζ(1−2it)A(−it,it))ϕ(tL2π)dt+Oε(X−12+ε), since Xd′(12+it)Xd(12+it)=log(π8∣d∣)−12Γ′Γ(14+a−it2)−12Γ′Γ(14+a+it2). This completes the proof.□ 4. Making the Ratios Conjecture’s prediction explicit In this section, we wish to compare the Ratios Conjecture’s prediction for D*(ϕ;X) in Theorem 1.3 with the result in Theorem 1.1. We recall the prediction obtained in (3.17), which for 1logX<c′<14 equals D*(ϕ;X)=1W*(X)∑*doddw(dX)12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r)+log(8∣d∣π)+12Γ′Γ(14+a−r2)+12Γ′Γ(14+a+r2)−2Xd(12+r)ζ(1−2r)A(−r,r))ϕ(iLr2π)dr+Oε(X−12+ε). (4.1) To begin, we focus on the first two terms in (4.1). Recalling (3.8), we write A(α,γ)=(1+21+α−21+γ3·21+2α+γ−2γ−21+α)∏p(1−1p1+α+γ)−1·(1−1(p+1)p1+2α−1(p+1)pα+γ). Using the fact that A(r,r)=1, we compute Aα(r,r)=2log23(21+2r−1)+∑plogp(p+1)(p1+2r−1). Furthermore, we have that ζ′(1+2r)ζ(1+2r)=−∑plogpp1+2r−1. We now have the following result. Lemma 4.1 Let ε>0and suppose that 1logX<c′<14. Then 12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r))ϕ(iLr2π)dr=−2L∑p>2j≥1logppj(1+1p)−1ϕ^(2jlogpL). (4.2) Remark 4.2 In [8, Lemma 3.7], it was shown that the right-hand side of (4.2) is asymptotic to −ϕ(0)/2 as X→∞. The Katz–Sarnak prediction (1.2) is obtained by combining this term with the main terms occurring in Lemmas 4.3 and 4.6. Proof of Lemma 4.1 We have that 12πi∫(c′)(2ζ′(1+2r)ζ(1+2r)+2Aα(r,r))ϕ(iLr2π)dr=1πi∫(c′)(−∑plogpp1+2r−1+2log23(21+2r−1)+∑plogp(p+1)(p1+2r−1))ϕ(iLr2π)dr=1πi∫(c′)(2log23∑j=1∞12(1+2r)j−∑plogp∑j=1∞1p(1+2r)j+∑plogp(p+1)∑j=1∞1p(1+2r)j)ϕ(iLr2π)dr=1πi∫(c′)(2log23∑j=1∞12(1+2r)j−∑pplogpp+1∑j=1∞1p(1+2r)j)ϕ(iLr2π)dr. (4.3) Making the substitution u=−iLr2π, we have that (4.3) becomes 2L∫C′(2log23∑j=1∞e−(log2)(4πiujL)2j−∑pplogpp+1∑j=1∞e−(logp)(4πiujL)pj)ϕ(u)du, (4.4) where C′ denotes the horizontal line I(u)=−Lc′2π. We note that the summations inside the integral over C′ in (4.4) converge absolutely and uniformly on compact subsets. Thus, we may interchange the order of integration and summation and (4.4) becomes 4log23L∑j=1∞12j∫C′ϕ(u)e−2πiu(2jlog2L)du−2L∑pplogpp+1∑j=1∞1pj∫C′ϕ(u)e−2πiu(2jlogpL)du. Next, we move the contour of integration from C′ to the line I(u)=0. Note that this is allowed since ϕ^ has compact support on R and the entire function ϕ(z)≔∫Rϕ^(x)e2πixzdx satisfies the inequality ∣ϕ(T+it)∣≤12π∣T∣∫R∣ϕ^′(x)∣max(1,exLc′)dx, uniformly for −Lc′2π≤t≤0, as T→±∞. Thus, (4.3) equals −2L∑p>2plogpp+1∑j=1∞1pjϕ^(2jlogpL), which concludes the proof.□ We now study the third, fourth and fifth terms in (4.1). For these terms, we can shift the line of integration to the imaginary axis, since the integrands are analytic in this region. We now give estimates for these shifted integrals. Lemma 4.3 Fix ε>0. We have 1W*(X)∑*doddw(dX)12π∫Rlog(8∣d∣π)ϕ(tL2π)dt=ϕ^(0)+ϕ^(0)L(log(24e)+2w^(0)∫0∞w(x)(logx)dx)+Oε,w(X−12+ε). Proof The result follows as in [8, Lemma 2.4] (see also [7, Lemma 2.8]).□ Lemma 4.4 We have 1W*(X)∑*doddw(dX)14π∫R(Γ′Γ(14+a−it2)+Γ′Γ(14+a+it2))ϕ(tL2π)dt=ϕ^(0)Llog(2−3e−γ)+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx. Proof We have that 1W*(X)∑*doddw(dX)14π∫R(Γ′Γ(14+a−it2)+Γ′Γ(14+a+it2))ϕ(tL2π)dt=14L∫R(Γ′Γ(14+πiuL)+Γ′Γ(14−πiuL)+Γ′Γ(34+πiuL)+Γ′Γ(34−πiuL))ϕ(u)du=(Γ′Γ(14)+Γ′Γ(34))ϕ^(0)2L+1L∫0∞e−x/2+e−3x/21−e−2x(ϕ^(0)−ϕ^(xL))dx, from [16, Lemma 12.14]. The result follows.□ The next lemma is required to evaluate the last term in (4.1). Lemma 4.5 Let ε>0and assume that 0≤R(r)≤12. Then we have the estimate 1W*(X)∑*doddw(dX)∣d∣−r=2w^(0)X−rMw(1−r)+Oε,w((∣I(r)∣+1)1384+εX−12−R(r)+ε). Proof We write the sum we are interested in as the Mellin integral S≔∑*doddw(dX)∣d∣−r=12πi∫(2)2s+r+12s+r+1ζ(s+r)ζ(2(s+r))XsMw(s)ds. We pull the contour of integration to the line R(s)=12−R(r)+ε. Then we have a contribution from the simple pole at s=1−r. Note that the restriction on R(r) ensures that we do not encounter the potential pole of Mw(s) at s=0. Hence, by the rapid decay of Mw on vertical lines, Bourgain’s subconvexity bound [1, Theorem 5] on ∣ζ(s+r)∣ and the boundedness of ∣ζ(2(s+r))−1∣, we have that S=43ζ(2)X1−rMw(1−r)+Oε,w((∣I(r)∣+1)1384+εX12−R(r)+ε). Similarly as in [7, Lemma 2.10], it can be shown that W*(X)=23ζ(2)Xw^(0)+Oε,w(X12+ε). We complete the proof by combining the last two estimates.□ We now give an asymptotic formula for the last term in (4.1). By a straightforward computation and by recalling the definitions of Xd and A, we apply the substitution r=2πiτ/L to obtain I≔−ζ(2)L∫C′(Γ(14−πiτL)Γ(14+πiτL)+Γ(34−πiτL)Γ(34+πiτL))(π8)2πiτL(1+2−24πiτL+14−24πiτL)ζ(1−4πiτL)ζ(2−4πiτL)ϕ(τ)×1W*(X)∑*doddw(dX)∣d∣−2πiτLdτ, (4.5) where C′ again denotes the horizontal line I(τ)=−Lc′2π. Lemma 4.6 We have the asymptotic formula I=ϕ(0)2−12∫−11ϕ^(τ)dτ+ϕ^(1)L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(L−2). Proof Let η>0 be small. We first change the contour of integration in I to the path C≔C0∪C1∪C2, where C0≔{I(τ)=0,∣R(τ)∣≥Lε},C1≔{I(τ)=0,η≤∣R(τ)∣≤Lε},C2≔{∣τ∣=η,I(τ)≤0}. For the part of the integral I over C0, we trivially bound the sum over d and use the rapid decay of ϕ to obtain the bound ≪ε1L∫C0max(log(∣τ∣+2),L∣τ∣−1)(∣τ∣+2)−3/ε2dτ≪∫LεLlog(∣τ∣+2)∣τ∣−1(∣τ∣+2)−3/ε2dτ+L−2/ε2≪L−3/ε(logL)2+L−2/ε2≪L−2/ε. On C1∪C2, we use Taylor expansions for each factor in the integrand of I, except the last in which we apply Lemma 4.5. This yields −ζ(2)L(Γ(14−πiτL)Γ(14+πiτL)+Γ(34−πiτL)Γ(34+πiτL))(π8)2πiτL×(1+2−24πiτL+14−24πiτL)ζ(1−4πiτL)ζ(2−4πiτL)ϕ(τ)W*(X)∑*doddw(dX)∣d∣−2πiτL=−1L(2+(γ+log8)4πiτL+O(∣τ∣2L2))(1+2πiτLlog(π8)+O(∣τ∣2L2))×(1+(2ζ′(2)ζ(2)−43log2)2πiτL+O(∣τ∣2L2))(−L4πiτ+γ+O(∣τ∣L))ϕ(τ)×2w^(0)(X−2πiτLMw(1−2πiτL)+Oε,w((∣I(iτ)∣L+1)1384+εX−12+ε))=12πiτ(1+2πiτL(−γ−log(243π)+2ζ′(2)ζ(2))+O(∣τ∣2L2))ϕ(τ)e−2πiτ×(1−2πiτlog(2πe)L+O(∣τ∣2L2))(1−Mw′(1)Mw(1)2πiτL+O(∣τ∣2L2))+Oε,w(X−12+ε)=12πiτ(1+2πiτL(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+O(∣τ∣2L2))ϕ(τ)e−2πiτ+Oε,w(X−12+ε). Hence I=I1+I2+Ow(L−2), where I1≔∫C1∪C212πiτϕ(τ)e−2πiτdτ and I2≔1L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))∫C1∪C2ϕ(τ)e−2πiτdτ. For the second integral I2, by the rapid decay of ϕ on the real line and the holomorphy of the integrand, we have that I2=1L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))∫Rϕ(τ)e−2πiτdτ+Oε(L−2/ε)=ϕ^(1)L(−γ−1−log(273)+2ζ′(2)ζ(2)−Mw′(1)Mw(1))+Oε(L−2/ε). Similarly, for the first integral I1, we find that I1=12πi∫C0∪C1∪C2e−2πiτϕ(τ)τdτ+Oε(L−2/ε)=J1+J2+Oε(L−2/ε), where J1≔12πi∫C0∪C1∪C2cos(2πτ)ϕ(τ)τdτ and J2≔−12π∫C0∪C1∪C2sin(2πτ)ϕ(τ)τdτ. We see that since the integrand in J1 is odd, the part of the integral on C0∪C1 is zero. Hence, J1=12πi∫C2cos(2πτ)ϕ(τ)τdτ, which by the residue theorem tends to ϕ(0)/2 as η tends to zero. As for the integral J2, we apply Plancherel’s identity. Since sin(2πτ)/τ is an entire function, J2=−∫Rsin(2πτ)2πτϕ(τ)dτ=−12∫RI[−1,1](τ)ϕ^(τ)dτ=−12∫−11ϕ^(τ)dτ, which coincides with the second term in the Katz–Sarnak prediction. Since all of our error terms are independent of η, we conclude the desired result.□ Proof of Theorem 1.4 The proof follows by combining Lemmas 4.1, 4.3, 4.4 and 4.6.□ Funding The first author was supported by an NSERC discovery grant. The third author was supported by a grant from the Swedish Research Council (Grant 2016-03759). Acknowledgements We thank Bruno Martin for helpful discussions. We also thank the referee for pointing our attention to the papers [13, 14] of Mason and Snaith. References 1 J. Bourgain , Decoupling, exponential sums and the Riemann zeta function , J. Amer. Math. Soc. 30 ( 2017 ), 205 – 224 . Google Scholar CrossRef Search ADS 2 J. B. Conrey , D. W. Farmer and M. R. Zirnbauer , Autocorrelation of ratios of L-functions , Commun. Number Theory Phys. 2 ( 2008 ), 593 – 636 . Google Scholar CrossRef Search ADS 3 J. B. Conrey and N. C. Snaith , Applications of the L-functions ratios conjectures , Proc. Lond. Math. Soc. (3) 94 ( 2007 ), 594 – 646 . Google Scholar CrossRef Search ADS 4 J. B. Conrey and N. C. Snaith , Correlations of eigenvalues and Riemann zeros , Commun. Number Theory Phys. 2 ( 2008 ), 477 – 536 . Google Scholar CrossRef Search ADS 5 H. Davenport , Multiplicative number theory, third edition, revised and with a preface by H. L. Montgomery, Graduate Texts in Mathematics Vol. 74, Springer-Verlag , New York , 2000 . 6 A. Entin , E. 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Kowalski , Analytic number theory, American Mathematical Society Colloquium Publications Vol. 53 , American Mathematical Society , Providence, RI , 2004 . 12 N. M. Katz and P. Sarnak , Zeroes of zeta functions and symmetry , Bull. Amer. Math. Soc. (N.S.) 36 ( 1999 ), 1 – 26 . Google Scholar CrossRef Search ADS 13 A. M. Mason and N. C. Snaith , Symplectic n-level densities with restricted support , Random Matrices Theory Appl. 5 ( 2016 ), 1650013 , 36 pp. Google Scholar CrossRef Search ADS 14 A. M. Mason and N. C. Snaith , Orthogonal and symplectic n-level densities , Mem. Amer. Math. Soc. 251 ( 2018 ), 93 pp . 15 S. J. Miller , A symplectic test of the L-functions ratios conjecture , Int. Math. Res. Not. IMRN2008 ( 2008 ), 36 pp . Art. IDrnm146. 16 H. L. Montgomery and R. C. Vaughan , Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics Vol. 97, Cambridge University Press , Cambridge , 2007 . 17 A. E. Özlük and C. Snyder , On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis , Acta Arith. 91 ( 1999 ), 209 – 228 . Google Scholar CrossRef Search ADS 18 M. Rubinstein , Low-lying zeros of L-functions and random matrix theory , Duke Math. J. 109 ( 2001 ), 147 – 181 . Google Scholar CrossRef Search ADS 19 Z. Rudnick , Traces of high powers of the Frobenius class in the hyperelliptic ensemble , Acta Arith. 143 ( 2010 ), 81 – 99 . Google Scholar CrossRef Search ADS 20 P. Sarnak , S. W. Shin and N. Templier , Families of L-functions and their symmetry, Proceedings of Simons Symposia, Families of Automorphic Forms and the Trace Formula, Springer-Verlag ( 2016 ), 531–578. 21 V. S. Vladimirov , Equations of mathematical physics, translated from the Russian by Audrey Littlewood, Pure and Applied Mathematics Vol. 3, Marcel Dekker, Inc , New York , 1971 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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The Quarterly Journal of MathematicsOxford University Press

Published: Apr 11, 2018

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