maass aw veforms and low-lying zeroes

MAASS WAVEFORMS AND LOW-LYING ZEROES

LEVENT ALPOGE, NADINE AMERSI, GEOFFREY IYER, OLEG LAZAREV,STEVEN J. MILLER,AND LIYANG ZHANG

ABSTRACT. The Katz-Sarnak Density Conjecture states that the behavior of zeros of a familyof L-functions near the central point (as the conductors tend tozero) agrees with the behaviorof eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Us-ing the Petersson formula, Iwaniec, Luo and Sarnak [ILS] proved that the behavior of zerosnear the central point of holomorphic cusp forms agrees withthe behavior of eigenvalues oforthogonal matrices for suitably restricted test functionsφ. We prove similar results for fam-ilies of cuspidal Maass forms, the other natural family ofGL2/Q L-functions. For suitableweight functions on the space of Maass forms, the limiting behavior agrees with the expectedorthogonal group. We prove this forsupp(φ) ⊆ (−3/2, 3/2) when the levelN tends toinfinity through the square-free numbers; if the level is fixed the support decreases to beingcontained in(−1, 1), though we still uniquely specify the symmetry type by computing the2-level density.

CONTENTS

1. Introduction 21.1. Background 21.2. Normalizations and Notation 21.3. Main Results 41.4. Outline of Arguments 72. Preliminaries for the Proofs 82.1. Calculating the averaged one-level density 82.2. Handling the Bessel integral 113. Proofs of the Main Theorems 123.1. Proof of Theorem 1.2 123.2. Proof of Theorem 1.3 123.3. Proof of Theorem 1.4 143.4. Proof of Theorem 1.5 16Appendix A. Contour integration 16Appendix B. An exponential sum identity 19

Date: May 30, 2013.2000Mathematics Subject Classification.11M26 (primary), 11M41, 15A52 (secondary).Key words and phrases.1-level density, Maass forms, low-lying zeros, Kuznetsov trace formula.We thank Gergely Harcos, Andrew Knightly, Stephen D. Millerand Peter Sarnak for helpful conversations

on an earlier version. This work was done at the 2011 and 2012 SMALL REUs at Williams College, funded byNSF GRANT DMS0850577 and Williams College; it is a pleasure to thank them for their support. The secondnamed author was also partially supported by the Mathematics Department of University College London, andthe fifth named author was partially supported by NSF grant DMS0970067.

1

2 ALPOGE, AMERSI, IYER, LAZAREV, MILLER, AND ZHANG

References 20

The fifth named author was a post-doc at Ohio State, and benefited enormously from manyconversations with Steve Rallis on the properties ofL-functions, as well as his mentoring andadvice. It is a pleasure to dedicate this paper to his memory.

1. INTRODUCTION

1.1. Background. The distribution of zeros ofL-functions plays an important role in numer-ous problems in number theory, from the distribution of primes [Con, Da] to the size of theclass groups of imaginary quadratic fields [CI, Go, GZ]. In the 1970s, Montgomery [Mon]observed that the pair correlation of the zeros ofζ(s), for suitable test functions, agrees withthat of the eigenvalues of the Gaussian Unitary Ensemble (GUE). This suggested a powerfulconnection between number theory and random matrix theory (see [FM, Ha] for some of thehistory), which was further supported by Odlyzko’s investigations [Od1, Od2] showing agree-ment between the spacings of zeros ofζ(s) and the eigenvalues of the GUE. Later studies byKatz and Sarnak [KaSa1, KaSa2] showed more care is needed. Specifically, although then-level correlations agree for suitable test functions [Hej,RS] and are the same for all classicalcompact groups, the behavior of the eigenvalues near 1 is different for unitary, symplectic andorthogonal matrices. This led to the Katz-Sarnak Density Conjecture, which states that thebehavior of zeros near the central point in a family ofL-functions (as the conductors tend toinfinity) agrees with the behavior of eigenvalues near 1 of a classical compact group (as thematrix size tends to infinity). For suitable test functions,this has been verified in many fami-lies, including Dirichlet characters, elliptic curves, cuspidal newforms, symmetric powers ofGL(2) L-functions, and certain families ofGL(4) andGL(6) L-functions; see for example[DM1, DM2, FI, Gao, Gü, HM, HR, ILS, KaSa2, Mil2, MilPe, OS, RR, Ro, Rub, Ya, Yo].This correspondence between zeros and eigenvalues allows us, at least conjecturally, to as-sign a definite symmetry type to each family ofL-functions (see [DM2, ShTe] for more onidentifying the symmetry type of a family).

For this work, the most important families studied to date are holomorphic cusp forms.Using the Petersson formula (and a delicate analysis of the resulting Bessel-Kloostermanterm), Iwaniec, Luo, and Sarnak [ILS] proved that the limiting behavior of the zeros nearthe central point of holomorphic cusp forms agrees with thatof the eigenvalues of orthogonalmatrices for suitably restricted test functions. In this paper we look at the otherGL2/Q familyof L-functions, Maass waveforms.

1.2. Normalizations and Notation. We quickly recall the basic properties of Maass forms(see [Iw2, IK, Liu, LiuYe] for details), and then review the 1-level density. We use the standardconventions. Specifically, byA ≪ B we mean|A| ≤ c|B| for a positive constantc. Similarly,A ≍ B meansA ≪ B andA ≫ B. We sete(x) := exp(2πix), and define the Fouriertransform off by

f(ξ) =

∫

R

f(x)e(−xξ)dx. (1.1)

MAASS WAVEFORMS AND LOW-LYING ZEROES 3

Let u be a Maass cusp form onΓ0(N), N squarefree, with Laplace eigenvalueλu =:(12+ itu)(

12− itu). Selberg’s3/16ths theorem implies that we may taketu ≥ 0 or tu ∈ [0, 1

4]i.

Next we Fourier expandu as follows:

u(z) = y1/2∑

n 6=0

an(u)Ks−1/2(2π|n|y)e(ny). (1.2)

Let

λn(u) :=an(u)

cosh(πtu)1/2. (1.3)

We normalizeu so thatλ1(u) = 1.TheL-function associated tou is

L(s, u) :=∏

p∤N

(1− λpp−s + χ0(p)p

−2s)−1, (1.4)

whereχ0 is the principal character moduloN . By work of Kim-Sarnak [KSa] towards theRamanujan conjecture for these forms, this product converges in the right half-planeRe s >1 + 7

64. Let

Λ(s, u) := N s/2π−sΓ

(s+ ǫ+ itu

2

)Γ

(s+ ǫ− itu

2

)L(s, u) (1.5)

be the completedL-function, whereu(−z) = (−1)ǫu(z), ǫ ∈ {0, 1} (that is, we may furtherdiagonalizez 7→ −z on the eigenspaces of the Laplacian). Letu

(− 1

Nz

)=: (−1)ǫ

′u(z),

ǫ′ ∈ {0, 1}, so(−1)ǫ′is the Atkin-Lehner eigenvalue ofu. Then

Λ(1− s, u) = (−1)ǫ+ǫ′+1Λ(s, u). (1.6)

For the remainder of the paperBN denotes an orthogonal basis of Maass cusp forms onΓ0(N), all normalized so thatλ1 = 1; thusBN is not orthonormal under the Petersson innerproduct on the space. Note we do not take a basis ofnewforms— that is, the delicate sievingout of oldforms as in [ILS] is not done. Of course such sievingis easy for e.g.N a prime bythe relevant Weyl law.

We use the notationAvg(A;w) to denote the average ofA overBN with each elementu ∈ BN given weightw(u). That is,

Avg(A;w) :=

∑u∈BN

A(u)w(u)∑u∈BN

w(u). (1.7)

Our main statistic for studying the low-lying zeros (i.e., the zeros near the central point) isthe 1-level density, which we quickly review; see [ILS] for agood introduction and formulasfor this statistic for the scaling limits of the various classical compact groups. Letφ be aneven Schwartz function such that the Fourier transformφ of φ has compact support; that is,

φ(y) =

∫ ∞

−∞φ(x)e−2πixydx (1.8)

and there is anη < ∞ such thatφ(y) = 0 for y outside(−η, η).


The1-level density ofL(s, u) is

D1(u, φ, R) =∑

ρ

φ

(logR

2πγ

), (1.9)

whereρ = 12+ iγ are the nontrivial zeros ofL(s, u), and logR is a rescaling parameter

related to the average log-conductor in the weighted family, whose choice is forced upon usby (2.5). Under GRH allγ are real and the zeros can be ordered; while GRH gives a niceinterpretation to the 1-level density, it is not needed for our purposes. Asφ is a Schwartzfunction, most of the contribution comes from the zeros nearthe central points = 1

2. The

different classical compact groups (unitary, symplectic,and orthogonal) have distinguishable1-level densities for arbitrarily small support; however,the 1-level densities for the even andodd orthogonal matrix ensembles are equal for test functions whose Fourier transforms aresupported in(−1, 1). There are two solutions to this issue. One possibility is toperforma more detailed analysis and “extend support”. The other is to study the 2-level density,which Miller [Mil1, Mil2] showed distinguishes the orthogonal ensembles for arbitrarily largesupport. For some of the families studied below we are able tocalculate the support beyond(−1, 1), and we may thus determine which of the orthogonal groups should be the symmetrygroup; for the other families our support is too limited and we instead study the 2-level density.

1.3. Main Results. Similar to how the harmonic weights facilitate applications of the Pe-tersson formula to average the Fourier coefficients of cuspidal newforms (see for instance[ILS, MilMo]), we introduce nice, even weight functions to smooth the sum over the Maassforms. In what follows, we consider the averaged one-level density weighted by two differentweight functions of “nice” analytic properties. LetH ∈ C∞ ((−1

4, 14

))be an even smooth

bump function of compact support on the real line, and letH be its Fourier transform. Wemay of course (by applying this construction to a square root— recall that the support of aconvolution is easily controlled) takeH ≥ 0. We may also takeH to have an orderK zero at0. Let

HT (r) := H( r

T

). (1.10)

This is essentially supported in a band of length≍ T about±T .Next, in the same way, leth ∈ C∞ ((−1

4, 14

))be even. We will also takeh to have an order

at least8 zero at0. Note that, by the same process as above, we may takeh(x) ≥ 0 for allx ∈ R and also (by Schwarz reflection)h(ix) ≥ 0 for all x ∈ R. Let T be a positive oddinteger. We let

hT (r) :=rTh(irT

)

sinh(πrT

) . (1.11)

This is the same test function used in [AM], and is essentially supported in a band of length≍ T about±T .

By trivially bounding the Fourier integral we observe that

H(x+ iy), h(x+ iy) ≪ exp

(π|y|2

). (1.12)


Hence

HT (ir) ≪ exp

(π|r|2T

), (1.13)

and, usingsinh(x) ≫ e|x|, we find

hT (r) ≪ exp

(−π|r|

4T

). (1.14)

These will both be useful in what follows.In one-level calculations we will take an even Schwartz functionφ such thatφ is supported

inside [−η, η]. We suppress the dependence of constants onh,H, φ andη (as these are allfixed), but notT or the levelN since one or both of these will be tending to infinity.

We weight each elementu ∈ BN by eitherHT (tu)||u||2 or hT (tu)

||u||2 , where

||u||2 = ||u||2Γ0(N)\h =1

[SL2(Z) : Γ0(N)]

∫

Γ0(N)\hu(z)

dxdy

y2(1.15)

is theL2-norm ofu on the modular curveY0(N), and, as before,λu = 14+ t2u is the Laplace

eigenvalue ofu. Recall that

[SL2(Z) : Γ0(N)] =: ν(N) =∏

p|N(p+ 1). (1.16)

The averaged one-level density may thus be written (we will see in (2.5) thatR ≍ T 2N isforced)

D1(BN , φ, R;w) := Avg

(D1(u, φ, R);

w(tu)

||u||2). (1.17)

The main question is to determine the behavior ofD1(BN , φ, R;w) as either the levelN orthe weight parameterT tends to infinity; specifically, one is generally interestedin the corre-sponding symmetry group. There are now several works [DM2, KaSa1, KaSa2, ShTe] whichsuggest ways to determine the symmetry group. For our family, they suggest the followingconjecture.

Conjecture 1.1. Let hT be as in(1.11), φ an even Schwartz function withφ of compactsupport, andR ≍ T 2N . Then

limR→∞

D1 (BN , φ, R;w) =

∫

R

φ(t)W1,O(t)dt, (1.18)

whereW1,O := 1+ 12δ0. In other words, the symmetry group associated to the familyof Maass

cusp forms of levelN is orthogonal.

In [AM] the above conjecture is shown forN = 1, w = hT , and with the extra restrictionthatsupp(φ) ⊆ (−5

4, 54). Our first result is in the case where the level tends to infinity.

We takeN → ∞ (remember thatN must be squarefree), fixT,K, and putw = wT equalto hT orHT to get:

Theorem 1.2. Fix T andK and letR ≍ T 2N . LetH be an even, non-negative function withK zeros at 0 and Fourier transformH ∈ C∞ ((−1

4, 14

)), and leth be an even function with 8

zeros at 0 andh ∈ C∞ ((−14, 14

)). Let the weightsw = wT be eitherHT or hT , where these


are the functions given by(1.10)and(1.11), respectively. Letφ be an even Schwartz functionwith supp(φ) ⊆ (−3

2, 32). Then

limN→∞

N squarefree

D1(BN , φ, R;wT ) =

∫

R

φ(t)W1,O(t)dt. (1.19)

Notice that the support in Theorem 1.2 exceeds(−1, 1), and thus we have uniquely speci-fied which orthogonal group is the symmetry group of the family.

Next we investigate the case whereN is fixed andT tends to infinity through odd values.For ease of exposition we takeN = 1.

Theorem 1.3. Let h be an even function with 8 zeros at 0 andh ∈ C∞ ((−14, 14

)), and

definehT as in(1.11). Letφ be an even Schwartz function withsupp(φ) ⊆ (−1, 1), and takeR ≍ T 2N . Then

limT→∞T odd

D1(B1, φ, R; hT ) =

∫

R

φ(t)W1,O(t)dt. (1.20)

We also get a similar, though slightly weaker, result for theweight functionw = HT

if we allow K = ordz=0H(z) to vary. For the argument given we invoke the work of [ILS]twice, since we reduce the Kloosterman-Bessel term of the Kuznetsov trace formula to sum ofKloosterman terms arising in the Petersson trace formula. Since [ILS] use GRH (specifically,for Dirichlet L-functions andL-functions associated to symmetric squares of holomorphiccusp forms), we must, too.

Theorem 1.4. Assume GRH for DirichletL-functions and symmetric squares of holomorphiccusp forms of level 1. LetH be an even, non-negative function withK zeros at 0 and FouriertransformH ∈ C∞ ((−1

4, 14

)), and let the weightsw = wT beHT , which is given by(1.10).

Letφ be an even Schwartz function withsupp(φ) ⊆ (−1+ 15+2K

, 1− 15+2K

). TakeR ≍ T 2N .Then

limT→∞

D1(B1, φ, R;HT ) =

∫

R

φ(t)W1,O(t)dt. (1.21)

Notice the support in Theorems 1.3 and 1.4 is too small to uniquely determine which or-thogonal symmetry is present (this is because the one-leveldensities of the orthogonal flavoursall agree inside(−1, 1)). At the cost of more technical arguments, Alpoge and Miller[AM]are able to extend the support beyond(−1, 1) when weighting byhT , thereby determining thesymmetry group to be orthogonal. In this work we instead compute the 2-level density, whichprovides a second proof that the symmetry type of the family of Maass cusp forms onSL2(Z)is orthogonal. The 2-level density is defined in (1.24). As any support for the 2-level densitysuffices to uniquely determine the symmetry group, we do not worry about obtaining optimalresults.

Theorem 1.5. LetwT equalhT or HT , R ≍ T 2, and let

N (−1) :=1

∑u∈B1

wT (tu)||uu||2

∑

u:(−1)ǫ+ǫ′+1=−1

wT (tu)

||u||2 (1.22)


be the weighted percentage of Maass forms inB1 with odd functional equation. Write

D2(φ1, φ2, R;wT ) := Avg

(D2(u, φ1, φ2, R);

wT (tu)

||u||2), (1.23)

with

D2(u, φ1, φ2, R) :=∑

i 6=±j

φ1

(logR

2πγi

)φ2

(logR

2πγj

)(1.24)

= D1(u, φ1, R)D1(u, φ2, R)− 2D1(u, φ1φ2, R) + δ(−1)ǫ+ǫ′+1,−1φ1(0)φ2(0),(1.25)

the average 2-level density of the weighted family of level 1Maass cusp forms, and the 2-leveldensity ofu ∈ B1, respectively. Letf ∗ g denote the convolution off andg. Then, forǫ ≪ 1

andφ1, φ2 even Schwartz functions supported in(−ǫ, ǫ),

limT→∞

D2(φ1, φ2, R;wT ) =

(φ1(0)

2+ φ1(0)

)(φ2(0)

2+ φ2(0)

)

+ 2

∫

R

|x|φ1(x)φ2(x)dx

− (1−N (−1))φ1(0)φ2(0)− 2(φ1 ∗ φ2)(0), (1.26)

agreeing with the 2-level density of the scaling limit of an orthogonal ensemble with propor-tions ofN (−1) SO(odd) matrices and1−N (−1) SO(even) matrices.

A similar result holds forBN — all the calculations will be standard given our work on theone-level densities.

1.4. Outline of Arguments. By a routine application of the explicit formula we immediatelyreduce the problem to studying averages of Hecke eigenvalues over the space of Maass cuspforms of levelN . For this we apply the Kuznetsov trace formula, as found in [KL]. We arequickly reduced to studying a term of shape

ν(N)∑

c≥1

S(m, 1; cN)

cN

∫

R

J2ir

(4π

√m

cN

)rwT (r)

cosh(πr)dr. (1.27)

In all cases the idea is to move the contour fromR to R − iY with Y → ∞. The propertiesof the weightshT or HT ensure that the integral along the moving line vanishes in the limit,so all that is left in place of the integral is the sum over poles, of shape (up to negligible errorin the case ofhT , which also has poles of its own)

∑

k≥0

(−1)kJ2k+1

(4π

√m

cN

)(2k + 1)wT

(2k + 1

2i

). (1.28)

Now theN → ∞ limit is very easy to take, as all the Bessel functions involved have zeroesat0. So the term does not contribute (the total mass is of orderT 2ν(N), cancelling theν(N)out in front). With some care we arrive at Theorem 1.2.

If instead we takeN = 1 andwT = HT , then, by standard bounds on Bessel functions,J2k+1(

4π√m

c) is very small fork larger than≍

√mc

. For us√m will always be bounded in


size by something that is≍ T η. Thus fork smaller than this range, the Bessel term is stillcontrolled but not too small. It is the term

wT

(2k + 1

2i

)≍ H

(2k + 1

2Ti

)≪(k

T

)K

(1.29)

that is small. Upon taking

K ≫ 1

1− η(1.30)

it is in fact small enough to bound trivially. For slightly larger support we instead appeal tothe bounds of [ILS] on sums of Kloosterman sums, which are derived from assuming GRHfor Dirichlet L-functions. This gives Theorem 1.4. In fact, the expressionwe get is exactly aweighted sum of terms appearing in [ILS] from the Kloosterman terms of Petersson formulas.It would be interesting to find a conceptual explanation for this.

The proof of Theorem 1.3 is a simplified version of the argument given in [AM], exceptconsiderably shortened — instead of delicate analysis of exponential sums, we just use Euler-Maclaurin summation, though as one would expect our supportis thus smaller than that in[AM].

The proof of Theorem 1.5 follows from the previous results and another application of theKuznetsov formula, this time to the inner product ofTpℓ with Tqℓ′ with p, q primes.

2. PRELIMINARIES FOR THE PROOFS

In this section we compute and analyze some expansions and resulting expressions that areuseful in the proofs of our main theorems. We start in §2.1 by using the explicit formula torelate the sum over zeros to sums over the Hecke eigenvalues of the associated cusp forms.The weights and normalizations are chosen to facilitate applying the Kuznetsov trace formulato these sums, which we do. After trivially handling severalof the resulting terms, in §2.2 weanalyze the Bessel function integral that arises. We then use these results in §3 to prove thestated theorems.

2.1. Calculating the averaged one-level density. We first quickly review the computationof the explicit formula; see [ILS, RS] for details. Letu ∈ BN , and for an even Schwartz testfunctionφ set

Φ(s) := φ

((s− 1

2

)logR

2πi

). (2.1)

Consider ∫

σ= 32

Φ(s)Λ′

Λ(s, u)ds. (2.2)

By moving the integration toσ = −12

and applying the functional equation, we find that

2

∫

σ= 32

Λ′

Λ(s, u)Φ(s)ds = D1(u, φ) (2.3)

(use the rapid decay ofφ along horizontal lines and Phragmen-Lindelöf to justify the shift).After expanding the logarithmic derivative out in the usualway, applying the Kim-Sarnak


bound, and noticing thatλp2 = λ2p − χ0(p) for χ0 the principal character moduloN , this

equality simplifies to

D1(u, φ) =φ(0)

2+ φ(0)

(logN + log(1 + t2u)

logR

)+O

(log logR + log logN

logR

)

+ 2

2∑

ℓ=1

∑

p

λpℓ log p

pℓ2 logR

φ

(ℓ log p

logR

). (2.4)

Thus, ifwT is essentially supported on≍ T (as arehT andHT ), the averaged one-leveldensity is (since||u|| ≍ 1 under our normalizations, by [Smi])

D1(BN , φ, R;wT ) =φ(0)

2+ φ(0)

(log(T 2N)

logR

)+O

(log logR + log logN

logR

)

+ 2

2∑

ℓ=1

∑

p

log p

pℓ2 logR

φ

(ℓ log p

logR

)Avg(λpℓ;wT ). (2.5)

Notice the above computation tells us the correct scaling touse isR ≍ T 2N .The difficulty is in determining the averages over Hecke eigenvalues. For this we use the

Kuznetsov formula forBN . LetwT equalhT orHT .

Theorem 2.1 (Kuznetsov trace formula (see [KL], page 86)). Letm ∈ Z+. Then

∑

u∈BN

λm(u)

||u||2 wT (tu) =δm,1ν(N)

π2

∫

R

rwT (t) tanh(πr)dr

− 1

π

∑

(ip)p|N∈{0,1}ω(N)

∫

R

σir(m, (ip))σir(1, (ip))mirwT (r)

||(ip)||2|ζ(1 + 2ir)|2 dr

+2i

π

ν(N)

N

∑

c≥1

S(m, 1;Nc)

c

∫

R

J2ir

(4π

√m

Nc

)rwT (r)

cosh(πr)dr, (2.6)

whereS is the usual Kloosterman sum,(ip)p|N runs through all0 ≤ ip ≤ 1 with p rangingover the prime factors ofN ,

σir(a, (ip)) :=

∏

p|Npip

−1−2ir∑

d|a

χ0(d mod∏

p|N p1−ip)

d2ir

∑

f∈(Z/∏

p|N pipZ)×

e

(af

d∏

p|N pip

),

(2.7)

J is the usual Bessel function, and

||(ip)||2 =∏

p|N

p1−ip

1 + p=

N

ν(N)∏

p|N pip. (2.8)

In our applications we always have(m,N) = 1 since we only takem = 1, p, or p2,which means that the contribution from the principal character in the definition ofσir may be


ignored. Also the inner sum inσir(a, (ip)) is of the form

∑

ξ∈(Z/nZ)×e

(ξ

n

)= µ(n) ≪ 1. (2.9)

Hence, bounding trivially and noting that ourm have at most three divisors, we find

σir(a, (ip)) ≪

∏

p|Npip

−1

. (2.10)

Also, by work of de la Vallee Poussin on the prime number theorem,ζ(1 + 2ir) ≫ log(2 +|r|)−1. Hence the second term in (2.6), the Eisenstein contribution, is

≪ ν(N)

N

∑

(ip)p|N

1∏p|N pip

∫

R

wT (r) log(2 + |r|)dr

≪ ν(N)T log T

N

∏

p|N1 +

1

p

=

(ν(N)

N

)2

T log T. (2.11)

In our applications we will always divide these expressionsby the corresponding expres-sion withm = 1, which gives the total mass of the family (“the denominator”in the sequel).We will see that it is of order≍ T 2ν(N) (see Corollary 2.4). Hence, since it will be dividedby something of order≍ T 2ν(N), the Eisenstein contribution is thus negligible forN or Tlarge.

Note that the diagonal term (that is, the first term of (2.6), with m = 1) is

ν(N)

∫

R

rwT (r) tanh(πr)dr ≍ T 2ν(N). (2.12)

Hence to show the claim about the total mass it suffices to bound the last term of (2.6) in thecase ofm = 1.

We have therefore reduced the computation of the weighted 1-level density to understand-ing the “Kloosterman-Bessel” terms. We isolate this resultbelow.

Lemma 2.2. If m = 1, p or p2 is coprime toN andwT equalshT or HT , then

∑

u∈BN

λm(u)

||u||2 wT (tu) = δm,1 ·(≍ T 2N

)

+O

((ν(N)

N

)2

T log T

)

+2i

π

ν(N)

N

∑

c≥1

S(m, 1;Nc)

c

∫

R

J2ir

(4π

√m

Nc

)rwT (r)

cosh(πr)dr, (2.13)

whereδm,1 · (≍ T 2N) is the product of a term on the order ofT 2N with Kronecker’s delta.


2.2. Handling the Bessel integral. As in [AM], the technical heart of the analysis of theKuznetsov formula is the following claim, which relies on the analytic properties ofhT andHT (see Appendix A for a proof).

Proposition 2.3. LetT be an odd integer. LetX ≤ T . LetwT equalhT or HT , where theseare the weight functions from Theorems 1.2 through 1.5. Then

∫

R

J2ir(X)rwT (r)

cosh(πr)dr = c1

∑

k≥0

(−1)kJ2k+1(X)(2k + 1)wT

((k +

1

2

)i

)

[+c2T

2∑

k≥1

(−1)kJ2kT (X)k2h(k)

](2.14)

= c1∑

k≥0

(−1)kJ2k+1(X)(2k + 1)wT

((k +

1

2

)i

)

[+O

(Xe−c3T

)], (2.15)

wherec1, c2, and c3 are constants independent ofX andT , and the terms in brackets areincluded if and only ifwT = hT .

Since Bessel functions of integer order are much better-studied objects than those of purelyimaginary order, this is a useful reduction. The calculation also realizes the Kloostermanterm in the Kuznetsov formula as a sort of average (though the“weight function” is growingexponentially in the case ofHT ) of Kloosterman terms arising in Petersson formulas over alleven weights.

For what follows we quickly note the following corollary, which determines the size of thedenominator (total mass) mentioned above.

Corollary 2.4. LetwT equalhT or HT (as above). Then

∑

u∈BN

wT (tu)

||u||2 ≍ T 2ν(N). (2.16)

Proof. It suffices to show that

ν(N)

N

∑

c≥1

S(1, 1; c)

c

∑

k≥0

(−1)kJ2k+1

(4π

cN

)(2k + 1)wT

((k +

1

2

)i

)≪ ν(N)T

N.

(2.17)

To see this, we bound trivially by using

Jk(x) ≪ (x/2)k

k!, (2.18)

and

sin

(2k + 1

2Tπ

)≫ T−1 (2.19)

in the case ofwT = hT . �


3. PROOFS OF THEMAIN THEOREMS

Using the results from the previous section, we can now proveour main theorems. Allarguments begin with the following reductions. We first use (2.5) to reduce the determinationof the 1-level density to that of sums of the weighted averages of λp andλp2. We then usethe Kuznetsov trace formula (Theorem 2.1) to analyze these sums. By Lemma 2.2 we arereduced to bounding off the contribution from the Kloosterman-Bessel term, to which weapply Proposition 2.3. Now we move to each of the cases.

3.1. Proof of Theorem 1.2. It suffices to study

S :=

2∑

ℓ=1

∑

p

2 log p

pℓ2 logR

φ

(ℓ log p

logR

)ν(N)

N

∑

c≥1

S(pℓ, 1; c)

c

·∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

cN

)(2k + 1)wT

(2k + 1

2i

), (3.1)

and boundS by something growing strictly slower thanν(N). This is because we get todivide this term by the total mass, which by Corollary 2.4 is of the orderT 2ν(N). As T isfixed, we are dividing by a quantity on the order ofν(N).

Bounding trivially, we find

S ≪ ν(N)

N

∑

c≥1

1

c

∑

k≥0

(2π)2k+1wT

(2k+12

i)

(2k)!

2∑

ℓ=1

∑

pℓ/2≤Rη/2

2 log p

logR(cN)

12+ǫ−2k−1pkℓ

≪ ν(N)

N32−ǫ

∑

k≥0

(2π)2k

(2k)!wT

(2k + 1

2i

)Rη(k+1)

N2k logR

≪ ν(N)Nη− 3

2+ǫT 2η

logN + log TeOT (T 2ηNη−2). (3.2)

As T is fixed, the above is negligible forη < 32, which completes the proof. 2

3.2. Proof of Theorem 1.3. It suffices to study

S :=

2∑

ℓ=1

∑

p

log p

pℓ2 log T

φ

(ℓ log p

2 log T

)∑

c≥1

S(pℓ, 1; c)

c

·∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

c

)(2k + 1)hT

(2k + 1

2i

)(3.3)

and boundS by something growing strictly slower thanT 2. This is becauseN is fixed, so byCorollary 2.4 the denominator that occurs in the weighted averages is on the order ofT 2.


In [AM] (see their (3.8) to (3.18) — for the convenience of thereader, this argument isreproduced in Appendix B), it is proved that

∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

c

)(2k + 1)hT

(2k + 1

2i

)

= c8T∑

|α|<T2

e(Y sin

(παT

))˜h

(πY

Tcos(παT

))+O(Y )

=: c8TSh(Y ) +O(Y ), (3.4)

where˜h(x) := x2h(x) andY := 2pℓ/2

c. We apply the Euler-Maclaurin summation formula to

the first term, yielding

Sh(Y ) =

∫ T2

−T2

e(Y sin

(παT

))˜h

(πY

Tcos(παT

))dα

+M∑

k=2

Bk

k!

(e(Y sin

(παT

))˜h

(πY

Tcos(παT

)))(k)∣∣∣∣∣

T2

−T2

+ O

(∫ T2

−T2

∣∣∣∣∣

(e(Y sin

(παT

))˜h

(πY

Tcos(παT

)))(M)∣∣∣∣∣ dα

). (3.5)

In differentiating the expression

e(Y sin

(παT

))˜h

(πY

Tcos(παT

))(3.6)

k times, the worst case is when we differentiate the exponential every single time and pickup a factor of

(YT

)k; otherwise we gain at least one factor ofT (remember thatY should be

thought of as orderT η). Hence we may bound the error term by

∫ T2

−T2

∣∣∣∣∣

(e(Y sin

(παT

))˜h

(πY

Tcos(παT

)))(M)∣∣∣∣∣ dα ≪

(Y

T

)M

T. (3.7)

TakingM ≥ 1 + 11−η

, the error term is thusO(YT

).

Next, by the same analysis, in the second term of (3.5) we either differentiate the exponen-tial every single time, or we gain a factor ofT from differentiatingh or one of thecos

(παT

)’s

produced from differentiating the exponential. Thus, since we differentiate at least twice, allbut one term in thek-fold derivative is bounded byY

T 2 . The last remaining term, obtained by

differentiating the exponentialk times, vanishes because˜h has a zero at0.Hence it remains to bound the first term of (3.5). This we do by integrating by parts, via

∫eφ(x)f(x)dx = − 1

2πi

∫eφ(x)

(f(x)

φ′(x)

)′dx. (3.8)


We get∫ T

2

−T2

e(Y sin

(παT

))˜h

(πY

Tcos(παT

))dα

= c10Y

T 2

∫ T2

−T2

e(Y sin

(παT

))h′(πY

Tcos(παT

))sin(πα

t

)dα

≪ Y

T. (3.9)

Hence we obtain the bound

Sh(Y ) ≪ Y

T. (3.10)

Thus

∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

c

)(2k + 1)hT

(2k + 1

2i

)≪ p

ℓ2

c. (3.11)

That is, this tells us that

2∑

ℓ=1

∑

p

log p

pℓ2 log T

φ

(ℓ log p

2 log T

)∑

c≥1

S(pℓ, 1; c)

c

[∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

c

)(2k + 1)hT

(2k + 1

2i

)]

≪2∑

ℓ=1

∑

p

log p

pℓ2 log T

∣∣∣∣φ(ℓ log p

2 log T

)∣∣∣∣∑

c≥1

c−12+ǫ

[p

ℓ2

c

]

≪2∑

ℓ=1

∑

pℓ2 ≤T η

log p

log T

≪ T 2η

log T. (3.12)

Forη < 1 this is negligible upon division by the total mass (which is of orderT 2), completingthe proof. 2

3.3. Proof of Theorem 1.4. Before proceeding it bears repeating that the same limitingsup-port asK gets large (namely,(−1, 1)) can be achieved by just trivially bounding as above(that is, without exploiting cancellation in sums of Kloosterman sums), but we present herean argument connecting the Kuznetsov formula to the Petersson formula as studied in [ILS]instead.

It suffices to study

S :=2∑

ℓ=1

∑

p

log p

pℓ2 log T

φ

(ℓ log p

2 log T

)∑

c≥1

S(pℓ, 1; c)

c

·∑

k≥0

(−1)kJ2k+1

(4πp

ℓ2

c

)(2k + 1)HT

(2k + 1

2i

)(3.13)


and boundS by something growing strictly slower thanT 2. Let

Q∗k(m; c) := 2πik

∑

p

S(p, 1; c)Jk−1

(4πm

√p

c

)φ

(log p

logR

)2 log p√p logR

, (3.14)

exactly as in [ILS]. Then this simplifies to (dropping constants)

∑

c≥1

c−1∑

k≥0

(2k + 1)H

(2k + 1

2Ti

)Q∗

2k+2(1; c) (3.15)

if we ignore theℓ = 2 term, which is insignificant by e.g. GRH for symmetric squareL-functions onGL3/Q (as in [ILS]).

In Sections 6 and 7 of [ILS] they prove

Theorem 3.1. Assume GRH for all DirichletL-functions. Then

Q∗k(m; c) ≪ γk(z)mT ηkǫ(log(2c))−2 (3.16)

where

γk(z) :=

{2−k k ≥ 3z

k−1/2 else.(3.17)

and

z :=4πT η

c. (3.18)

Hence the sum overc now converges with no problem, and we may ignore it. What remainsis

T η∑

k≥0

(2k + 1)H

(2k + 1

2Ti

)γk(z)k

ǫ

= T η

(∑

0≤k ≪ T η

kǫ+ 12H

(2k + 1

2Ti

)+∑

T η ≪ k

k1+ǫH(2k+12T

i)

2k

). (3.19)

The second term in parentheses poses no problem. For the firstterm, usingH(x) ≪ xK ,we see that the above is bounded by

T η∑

0≤k ≪ T η

kǫ+ 12+KT−K ≪ T

52η−(1−η)K . (3.20)

Thus we need

η <2 +K52+K

= 1− 1

5 + 2K. (3.21)

Again, by takingK even larger we could have just trivially bounded throughoutand notinvoked [ILS] or GRH, but the connection noted above may be ofindependent interest.


3.4. Proof of Theorem 1.5. By definition,

D2(u, φ1, φ2, R) = D1(u, φ1, R)D1(u, φ2, R)− 2D1(u, φ1φ2, R) + δ(−1)ǫ+ǫ′+1,−1φ1(0)φ2(0).(3.22)

Averaging, we see that forφ1 and φ2 of sufficiently small support (actually(−1

2, 12

)would

work fine), given our results above on one-level densities, up to negligible error

D2(φ1, φ2, R;wT ) := Avg

(D1(u, φ1, R)D1(φ2, u);

wT (tu)

||u||2)

− (1−N (−1))φ1(0)φ2(0)− 2φ1 ∗ φ2(0). (3.23)

Now

D1(u, φ1, R)D1(φ2, u) =

(φ1(0)

log(1 + t2u)

logR+

φ1(0)

2−

2∑

ℓ=1

∑

p

2 log p

pℓ2 logR

φ1

(ℓ log p

logR

))

·(φ2(0)

log(1 + t2u)

logR+

φ2(0)

2−

2∑

ℓ=1

∑

p

2 log p

pℓ2 logR

φ2

(ℓ log p

logR

)).

(3.24)

SincewT is supported essentially aroundtu ≍ T , the log(1 + t2u) terms are all, up tonegligible error, approximatelylogR (again we invoke the bound of [Smi] on theL2-normsoccurring in the denominator). Also by an application of Kuznetsov, now with the innerproduct of two Hecke operators (namelyTpℓ andTqℓ′ for p, q primes and0 ≤ ℓ, ℓ′ ≤ 2 — thisuses the results on the Kloosterman-Bessel term established above), we see that the resultingaverage is, up to negligible error,

Avg

(D1(u, φ1, R)D1(φ2, u);

wT (tu)

||u||2)

=

(φ1(0) +

φ1(0)

2

)(φ2(0) +

φ2(0)

2

)

+

2∑

ℓ=1

∑

p

4 log2 p

pℓ log2Rφ1

(ℓ log p

logR

)φ2

(ℓ log p

logR

).

(3.25)

That is, only the diagonal termspℓ = qℓ′

matter. Now partial summation (and the primenumber theorem, as usual) finishes the calculation.

APPENDIX A. CONTOUR INTEGRATION

We prove Proposition 2.3 below. We restate it for the reader’s convenience.


Proposition A.1. LetT be an odd integer andX ≤ T . LetwT equalhT or HT , where theseare the weight functions from Theorems 1.2 to 1.5. Then

∫

R

J2ir(X)rwT (r)

cosh(πr)dr = c1

∑

k≥0

(−1)kJ2k+1(X)(2k + 1)wT

((k +

1

2

)i

)

[+ c2T

2∑

k≥1

(−1)kJ2kT (X)k2h(k)

](A.1)

= c1∑

k≥0

(−1)kJ2k+1(X)(2k + 1)wT

((k +

1

2

)i

)

[+O

(Xe−c3T

)], (A.2)

wherec1, c2, and c3 are constants independent ofX andT , and the terms in brackets areincluded if and only ifwT = hT .

Proof. In the proof below bracketed terms are present if and only ifwT = hT .Recall that

Jα(2x) =∑

m≥0

(−1)mx2m+α

m!Γ(m+ α + 1). (A.3)

By Stirling’s formula,Γ(m+ α+ 1) cosh(πr) ≫ |m+ 2ir + 1|m+1. Hence by the LebesgueDominated Convergence Theorem (remember thatwT (z) is of rapid decay as|Re z| → ∞)we may switch sum and integral to get

∫

R

J2ir(X)rwT (r)

cosh(πr)dr

=∑

m≥0

(−1)mx2m

m!

∫

R

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)dr, (A.4)

whereX =: 2x.Now we move the line of integration down toR− iR, R 6∈ Z+ 1

2(or TZ if wT = hT ). To

do this, we note the estimate (forA ≫ 1)∫

±A→±A−iR

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)dr ≪

∫

±A→±A−iR

x2R|r||m+ 2ir|−m|wT (r)|dr

≪T,R A−2, (A.5)

where again we have used the rapid decay ofwT along horizontal lines, andB → B − iRdenotes the vertical line fromB ∈ C to B − iR ∈ C. (Rapid decay also ensures the integralalongR− iR converges absolutely.)

Note that the integrand

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)(A.6)

has poles precisely atr ∈ −i(N + 12) = {−1

2i,−3

2i, . . .}, and, ifwT = hT , poles also at

r ∈ −iTZ+. The residue of the pole atr = −2k+12

i (k ≥ 0) is, up to an overall constant


independent ofk,

x2k+1

Γ(m+ 1 + (2k + 1))(−1)k(2k + 1)wT

(2k + 1

2i

). (A.7)

If wT = hT , the residue of the pole atr = −ikT (k ≥ 1) is, up to another overall constantindependent ofk,

x2kT

Γ(m+ 1 + (2kT ))k2T 2 h(k)

cos(πkT )=

x2kT

Γ(m+ 1 + (2kT ))(−1)kk2T 2h(k). (A.8)

Hence the sum of (A.4) becomes∑

m≥0

(−1)mx2m

m!

∫

R

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)dr

= c1∑

m≥0

(−1)mx2m

m!

(∑

0≤k≪R

x2k+1

Γ(m+ 1 + (2k + 1))(−1)k(2k + 1)wT

(2k + 1

2i

)

+

∫

R−iR

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)dr

[+c2

∑

m≥0

(−1)mx2m

m!

∑

0≤k≪R

x2kT

Γ(m+ 1 + (2kT ))(−1)kk2T 2h(k)

]).

(A.9)

Now we takeR → ∞. Note that∫

R−iR

x2irrwT (r)

Γ(m+ 1 + 2ir) cosh(πr)dr ≪

∫

R−iR

x2R|r||wT (r)||m+ 2ir + 1|−m−1−2Rdr

≪m x2ReπR2T R−m−1−2R, (A.10)

again by Stirling and rapid decay ofwT on horizontal lines (that is,wT (x+iy) ≪ (1+x)−4eπy2T

since bothh andH have all their derivatives supported in(−1

4, 14

)). This of course vanishes

asR → ∞.Hence we see that∫

R

J2ir(X)rwT (r)

cosh(πr)dr

= c1∑

m≥0

(−1)mx2m

m!

∑

k≥0

x2k+1

Γ(m+ 1 + (2k + 1))(−1)k(2k + 1)wT

(2k + 1

2i

)

[+c2

∑

m≥0

(−1)mx2m

m!

∑

k≥0

x2kT

Γ(m+ 1 + (2kT ))(−1)kk2T 2h(k)

]. (A.11)

Switching sums (via the exponential bounds onwT along the imaginary axis) and applyingJn(X) =

∑m≥0

(−1)mx2m+n

m!(m+n)!gives us the claimed calculation. For the bound on the bracketed

term, use

Jn(ny) ≪ynen

√1−y2

(1 +√

1− y2)n(A.12)


(see [AS], page 362) and bound trivially. �

APPENDIX B. AN EXPONENTIAL SUM IDENTITY

The following is the argument given in [AM], copied (essentially) verbatim.

Proposition B.1. SupposeX ≤ T . Then

SJ(X) := T∑

k≥0

(−1)kJ2k+1(X)˜h(2k+12T

)

sin(2k+12T

π)

= c8T∑

|α|<T2

e(Y sin

(παT

))˜g

(πY

Tcos(παT

))+O(Y ), (B.1)

wherec8 is some constant,X =: 2πY , and for anyf setf(x) := xf(x).

Proof. Observe thatk 7→ sin(πk2

)is supported only on the odd integers, and maps2k + 1 to

(−1)k. Hence, rewriting gives

SJ(X) = T∑

k≥0k 6∈2TZ

Jk(X)˜h

(k

2T

)sin(πk2

)

sin(πk2T

) . (B.2)

As

sin(πk2

)

sin(πk2T

) =e

πik2 − e−

πik2

eπik2T − e

πik2T

=

T−12∑

α=−(T−12 )

eπikαT (B.3)

whenk is not a multiple of2T , we find that

SJ(X) = T∑

|α|<T2

∑

k≥0k 6∈2TZ

e

(kα

2T

)Jk(X)˜h

(k

2T

). (B.4)

Observe that, since the sum overα is invariant underα 7→ −α (and it is non-zero only forkodd!), we may extend the sum overk to the entirety ofZ at the cost of a factor of 2 and ofreplacingh by

g(x) := sgn(x)h(x). (B.5)

Note thatg is as differentiable ash has zeroes at0, less one. That is to say,g decays like thereciprocal of a degreeordz=0 h(z)− 1 polynomial at∞. This will be crucial in what follows.

Next, we add back on the2TZ terms and obtain

1

2SJ(X) = T

∑

|α|<T2

∑

k∈Ze

(kα

2T

)Jk(X)˜g

(k

2T

)− T 2

∑

k∈ZJ2kT (X)k2h(k)

= T∑

|α|<T2

∑

k∈Ze

(kα

2T

)Jk(X)˜g

(k

2T

)+O

(Xe−c4T

)

=: VJ(X) +O(Xe−c4T

), (B.6)

where we have bounded the termT 2∑

k∈Z J2kT (X)k2h(k) trivially via Jn(2x) ≪ xn/n!.


Now we move to apply Poisson summation. WriteX =: 2πY . We apply the integralformula (fork ∈ Z)

Jk(2πx) =

∫ 12

− 12

e (kt− x sin(2πt)) dt (B.7)

and interchange sum and integral (via rapid decay ofg) to get that

VJ(X) = T∑

|α|<T2

∫ 12

− 12

(∑

k∈Ze

(kα

2T+ kt

)˜g

(k

2T

))e (−Y sin(2πt)) dt. (B.8)

By Poisson summation, (B.8) is just (interchanging sum and integral once more)

VJ(X) = T 2∑

|α|<T2

∑

k∈Z

∫ 12

− 12

g′′ (2T (t− k) + α) e (−Y sin(2πt)) dt

= c5T∑

|α|<T2

∫ ∞

−∞g′′(t)e

(Y sin

(πt

T+

πα

T

))dt

=: c5Wg(X). (B.9)

As

sin

(πt

T+

πα

T

)= sin

(παT

)+

πt

Tcos(παT

)− π2t2

T 2sin(παT

)+O

(t3

T 3

), (B.10)

we see that

Wg(X) = c6T∑

|α|<T2

e(Y sin

(παT

))∫ ∞

−∞g′′(t)e

(πY t

Tcos(παT

))dt

− c7π2Y

T

∑

|α|<T2

e(Y sin

(παT

))sin(παT

)∫ ∞

−∞t2g′′(t)e

(πY t

Tcos(παT

))dt

+O

(Y

T+

Y 2

T 2

)(B.11)

= c8T∑

|α|<T2

e(Y sin

(παT

))˜g

(πY

Tcos(παT

))

− c9Y

T

∑

|α|<T2

e(Y sin

(παT

))sin(παT

)˜g′′(πY

Tcos(παT

))

+O

(Y

T+

Y 2

T 2

). (B.12)

Now bound the second term trivially to get the claim. �

REFERENCES

[AILMZ] N. Amersi, G. Iyer, O. Lazarev, S. J. Miller and L. Zhang,Low-lying Zeros of Cuspidal Maass Forms,preprint.http://arxiv.org/abs/1111.6524.

[AM] L. Alpoge and S. J. Miller, Low lying zeroes of Maass formL-functions, preprint.http://arxiv.org/abs/1301.5702.

http://arxiv.org/abs/1111.6524

http://arxiv.org/abs/1301.5702


[AS] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs,andMathematical Tables, 9th printing, New York: Dover, 1972.

[Con] J. B. Conrey,L-Functions and random matrices. Pages 331–352 inMathematics unlimited — 2001and Beyond, Springer-Verlag, Berlin, 2001.

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250.

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