A Large Sieve Zero Density Estimate for Maass Cusp Forms

Paul Dunbar Lewis

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2017 © 2017 Paul Dunbar Lewis All rights reserved ABSTRACT

A Large Sieve Zero Density Estimate for Maass Cusp Forms

Paul Dunbar Lewis

The large sieve method has been used extensively, beginning with Bombieri in 1965, to provide bounds on the number of zeros of Dirichlet L-functions near the line σ = 1. Using the Kuznetsov trace formula and the work of Deshouillers and Iwaniec on Kloosterman sums, it is possible to derive large sieve inequalities for the Fourier coefficients of Maass cusp forms, which may then similarly be used to study the corresponding Hecke-Maass L-functions. Following an approach developed by Gallagher for Dirichlet L-functions, this thesis shows how the large sieve method may be used to prove a zero density estimate, averaged over the Laplace eigenvalues, for Maass cusp forms of weight zero for the congruence subgroup Γ0(q) for any positive integer q. Contents

Acknowledgements iii

Introduction 1

1 Preliminaries on Maass forms for Γ0(q) 10

1.1 Automorphic forms for GL(2) ...... 10

1.2 Maass Forms of Weight Zero ...... 13

1.3 Eisenstein Series and the Selberg Spectral Decomposition ...... 19

2 The Kuznetsov Trace Formula and Large Sieve Inequalities for Fourier Coefficients 21

2.1 Kloosterman Sums ...... 21

2.2 Kuznetsov Trace Formula ...... 29

2.3 Large Sieve Inequality for Maass Forms ...... 34

3 Zero Density Estimate for L-functions 38

3.1 Turán’s Method ...... 38

3.2 Exponential Sums with Fourier Coefficients ...... 43

3.3 Proof of the Main Theorem ...... 44

i Bibliography 47 Acknowledgements

First and foremost, I would like to thank Dorian Goldfeld for his invaluable guidance and suggest- ing the topic which led to this thesis. I also owe the utmost gratitude to Patrick Gallagher for laying the foundation for the work that appears here. I thank Alex Kontorovich, Chao Li, and Wei Zhang as well for their service on my committee.

My fellow graduate students at Columbia have been a constant source of joy and inspiration: shout-outs to Rob Castellano, Karsten Gimre, João Guerreiro, Andrea Heyman, Jordan Keller,

Rahul Krishna, Pak-Hin Lee, Vivek Pal, Sebastien Picard, Natasha Potashnik, Remy van Dobben de Bruyn, and Mike Wong.

Finally, I owe everything to my family for their unwavering love and support. Thanks to my parents, Laura Dunbar and Jim Lewis, and a special thanks to my grandmother Shirley Dunbar for adopting New York as a second home during my time here. I would not be who I am without all of you.

iii To all my loved ones

iv Introduction

The large sieve, first developed by Linnik [23] to study the distribution of quadratic nonresidues, has become an indispensable tool in analytic number theory. At its core, the large sieve is an analytic tool used to show cancellation in exponential sums. A simple formulation, following Davenport and Halberstam [5], is as follows: let an be an arbitrary sequence of complex numbers, and, for real α, let M∑+N S(α) = ane(nα), n=M+1 where e(θ) = e2πiθ. Note that S(α) is periodic with period 1, so the values depend only on the equivalence class of α in R/Z. Let α1, . . . , αR be a set of real numbers which are “δ-separated” in

R/Z in the sense that

||αr − αs|| ≥ δ when r ≠ s, and || · || denotes the distance to the nearest integer. Then the large sieve inequality of Davenport and Halberstam shows that

( ) ∑R 1 M∑+N |S(α )|2 ≪ N + |a |2. r δ n r=1 n=M+1

1 In fact, as proved by Montgomery and Vaughan [26], the implied constant above can be taken to

be 1, so the ≪ sign may be replaced with a ≤ sign.

This form of the large sieve found an application in a famous theorem proved independently by Bombieri [3] and Vinogradov [37, 38]:

Theorem 0.1 (Bombieri-Vinogradov). Let

∑ ψ(x; q, a) = Λ(n), n≤x n≡a mod q where Λ(n) is the Von Mangoldt function:

{ log p n = pk, p prime, k ≥ 1, Λ(n) = 0 otherwise.

≤ x1/2 Then for any positive constant A, there exists a positive constant B such that if Q (log x)B ,

∑ y −A max max ψ(y; q, a) − ≪ x(log x) . (a,q)=1 y≤x ϕ(q) q≤Q

This may be viewed as an averaged form of the Generalized Riemann Hypothesis (GRH) for

Dirichlet L-functions, as GRH would imply

y 1/2 2 max max ψ(y; q, a) − ≪ x log x, (a,q)=1 y≤x ϕ(q)

giving the theorem, but nothing stronger. That the theorem is as good as GRH in this averaged

context shows its power and utility.

2 The Bombieri-Vinogradov theorem is of particular interest recently as it played a key part in the work of Zhang [42] on bounded gaps between primes. In particular, the work of Goldston, Pintz, and Yildirim [14] shows that the Bombieri-Vinogradov theorem is, to quote the authors, “within a hair’s breadth” of establishing bounded gaps: if instead of requiring Q ≤ x1/2(log x)−B we could instead take Q = xϑ−ϵ for some ϑ > 1/2 and for all ϵ > 0, it would follow that

lim inf(pn+1 − pn) < ∞, n→∞

where pn denotes the nth prime. The Elliot-Halberstam conjecture [8] states that it is possible to take ϑ = 1 above. With this assumption, Maynard [25] was able to prove conditionally that

lim inf(pn+1 − pn) ≤ 12. n→∞

Zhang, while unable to break through the barrier of ϑ = 1/2 in full generality, does so in a limited way by restricting to moduli q which do not have large prime divisors, and this is enough to derive his celebrated result

7 lim inf(pn+1 − pn) < 7 × 10 . n→∞

All of these successes stem from the application of the large sieve method to bound sums over

Dirichlet characters. A typical example is the following bound ([10], Lemma 3):

M+N 2 M+N ∑ ∑∗ ∑ ∑ ≪ 2 | |2 anχ(n) (N + Q ) an . q≤Q χ mod q n=M+1 n=M+1

3 Here the asterisk means that the sum is taken over only primitive characters. A heuristic interpre-

tation of this type of large sieve inequality is that, on average, the Dirichlet characters behave like

random points on the unit circle. Indeed, a trivial application of the Cauchy-Schwarz inequality to

the above would result in an N 2 term on the right-hand side above, and the improvement from N 2

to N is exactly what would be expected if the χ(n) were replaced by uniform random variables.

This aligns closely with the philosophy of GRH, which can be viewed as asserting that, for each

individual χ, the sequence µ(n)χ(n), where µ is the Möbius function, behaves like a random walk

in a similar sense. Thus, it is intuitive that the large sieve inequality above could stand in for GRH

in an averaged setting.

The large sieve for Dirichlet characters leads not only to the Bombieri-Vinogradov theorem,

but also to results on the distribution of zeros of Dirichlet L-functions, providing more evidence

that large sieve methods can be helpful when bounds towards GRH are needed. In fact, Bombieri’s

proof of the theorem proceeds by way of the following zero density estimate:

Theorem 0.2 (Bombieri [3], Theorem 5). Let Q be a finite set of positive integers, and define

M = max q and D = max d(q), where d(q) is the number of divisors of q. Let N(α, T ; χ) denote q∈Q q∈Q the number of zeros s = σ + it of L(s, χ) in the rectangle α ≤ σ ≤ 1, |t| < T , and let τ(χ) be the ∑ Gauss sum χ(a)e(a/q) . Then a mod q

∑ ∑ 1 4(1−α) |τ(χ)|2N(α, T ; χ) ≪ DT (M 2 + MT ) (3−2α) log10(M + T ), ϕ(q) q∈Q χ mod q

1 ≤ ≤ ≥ uniformly with respect to Q, for 2 α 1, T 2.

Though subsequent proofs of the Bombieri-Vinogradov theorem by Gallagher [9] and Vaughan

4 [36] were able to dispense with the explicit step of bounding zeros, the large sieve continued to prove helpful in deriving even stronger zero density estimates than Bombieri’s. One of the first such estimates is the following result of Jutila, which improves on Bombieri’s result by a factor of

T :

Theorem 0.3 (Jutila, [17]). There exists a constant c such that

∑ ∑∗ 7 4 1−α c N(α, T ; χ) ≪ (Q T ) α log (Q + T ). q≤Q χ mod q

Jutila applied this result in [17] to prove a version of the Bombieri-Vinogradov theorem for short intervals.

Another such result is the following, which is the classical inspiration for the main theorem of this thesis:

Theorem 0.4 (Gallagher, [10]). Let N(α, T ; χ) denote the number of zeros s = σ + it of L(s, χ) in the rectangle α ≤ σ ≤ 1, |t| < T . Then there exists a positive constant c such that

∑ ∑∗ N(α, T ; χ) ≪ T c(1−α). q≤T χ mod q

This theorem found a striking application when Montgomery and Vaughan [27] applied it in their proof that the number of even numbers ≤ X which cannot be expressed as a sum of two primes is < X1−δ for an effective positive constant δ.

From the point of view of the Langlands program, the Dirichlet L-functions are the L-functions attached to automorphic representations for the group GL(1, AQ), and thus the simplest examples of a rich theory of automorphic L-functions for GL(n). Thus, given the effectiveness of the large

5 sieve in studying Dirichlet L-functions, it is natural to inquire as to whether similar methods could

also be used to study other families of L-functions. This thesis investigates the applications of the large sieve method to the study of the simplest L-functions beyond the Dirichlet L-functions; namely, those associated to cusp forms for GL(2).

To extend the large sieve method to GL(2), the Hecke eigenvalues of cusp forms, or relatedly, their Fourier coefficients, must play the role of the Dirichlet characters above, and one would hope that the large sieve method can show these numbers are pseudo-randomly distributed in a similar way. This introduces an additional difficulty: the values taken by Dirichlet characters are well- spaced on the unit circle, so the analytic large sieve inequality above can be used to study sums involving Dirichlet characters quite directly. This is not exactly the case, however, for Fourier coefficients of cusp forms. Thus, an additional step is needed to translate from sums involving

Fourier coefficients to more directly understandable exponential sums. One important tool for this step in the case of GL(2) is the trace formula developed by Bruggeman [4] and Kuznetsov [22], which relates the Fourier coefficients of cusp forms to Kloosterman sums. This process of applying the trace formula to derive large sieve inequalities was carried out extensively by Deshouillers and

Iwaniec [6], whose work is a significant inspiration for the results below.

One of the early applications of this method was actually a bound for the Riemann zeta function, demonstrating the connection between the study of higher-degree L-functions and more elementary applications:

2/3 Theorem 0.5 (Iwaniec [16]). Let T ≥ 2, T0 = T and ϵ > 0. Then

∫ ( ) T +T0 4 1 ϵ ζ + it dt ≪ T0T . T 2

6 The main result of this thesis synthesizes the large sieve results first developed by Deshouillers

and Iwaniec with the classical techniques of Gallagher used to prove Theorem 0.4. Combining

these two approaches leads to a new type of zero density estimate for GL(2) L-functions which is similar to, though weaker than, the result of Gallagher quoted above, thus demonstrating the potential strength of the large sieve method of Bombieri, Gallagher, and others, when combined with more modern techniques, to provide new insights into the distribution of zeros of more general

L-functions.

Before stating the theorem, we require some notation, all of which is explained in detail in

Chapter 1. Fix an integer q ≥ 1. Let {fj}j≥1 be an orthonormal basis (with respect to the Petersson

1 2 inner product) for the space of Maass cusp newforms for Γ0(q) such that ∆fj = ( 4 + tj )fj, where

∆ is the Laplacian, and assume each fj is an eigenfunction of all the Hecke operators Tn. Write ∑ √ | | fj(z) = aj(n) yKitj (2π n y)e(nx). Then the main theorem is as follows: n=0̸

Theorem 0.6. Let Nj(α, T ) be the number of zeros s = σ + it in the region α ≤ σ ≤ 1, |t| ≤ T of the function L(s, fj). Then there exists a positive constant c such that for all ϵ > 0 and α sufficiently close to 1, ∑ | |2 aj(1) c(1−α)+ϵ Nj(α, T ) ≪ T . cosh πtj |tj |≤T

Thus, in comparison to the averaging over the conductor q which takes place in the Bombieri-

Vinogradov theorem and the original result of Gallagher, the averaging in this setting is over the

1 2 Laplace eigenvalues 4 + t of Maass forms, with the conductor fixed. This averaging is natural

from the point of view of the Kuznetsov trace formula, which exploits the spectral decomposition

2 of the Laplace operator on L (Γ0(q)\h) for fixed q.

To put this estimate in context, the best known bound for N(α, T ) for a single Maass form is

7 of the form T c(1−α) for large c (Lemke Oliver and Thorner [28]). See also Tang [32] for the bound

T 2(1−α)/α(log T )C , giving an improved exponent of T at the cost of a power of log T .

The approach used here is not the only way of applying the large sieve concept to study higher-

degree L-functions, and a variety of alternative methods of averaging have been successfully car-

ried out as well. Duke and Kowalski [7] developed a large sieve inequality for automorphic forms

for GL(n) averaging over the conductor, leading to the following analogue of Linnik’s theorem

on the least quadratic nonresidue for elliptic curves (via the famous modularity theorem of Wiles

[40]):

Theorem 0.7 (Duke-Kowalski [7]). Let M(Q, α) be the maximal number of isogeny classes of

semistable elliptic curves over Q with conductor ≤ Q which for every prime p ≤ (log Q)α have a

fixed number of points (mod p). Then for any ϵ > 0,

8 +ϵ M(Q, α) ≪ Q α .

One possibility for generalizing the approach in this thesis is to consider the analogous prob-

lem for GL(3). This has become more possible recently thanks to the work of Goldfeld and Kon-

torovich [13] and Blomer [2] on the GL(3) Kuznetsov trace formula. In particular, Young [41]

used Blomer’s version of the trace formula, along with a study of the corresponding generalized

Kloosterman sums, to derive a large sieve inequality for the Hecke eigenvalues of Maass cusp

3 +ϵ forms for SL(3, Z). Unfortunately, Young’s inequality has a term of the form N 2 in place of

the N 1+ϵ in the GL(2) case, rendering the methods below ineffective. Nevertheless, it may still be

possible to obtain nontrivial zero density estimates from this work, and this provides an interesting

opportunity for future research.

8 The exposition below begins with a summary of the basic theory of Maass forms for congruence

subgroups Γ0(q) ⊆ SL(2, Z), including all the definitions, notations and main theorems which will be needed in the subsequent parts. Next is a summary and adaptation of the work of Deshouillers and Iwaniec [6], including the relevant facts on Kloosterman sums, the Kuznetsov trace formula, and the derivation of the large sieve inequality which will be used to prove the main theorem.

Finally, the results of the previous chapters come together with the methodology of Gallagher [10] in the final chapter to complete the proof of the main theorem.

9 Chapter 1

Preliminaries on Maass forms for Γ0(q)

1.1 Automorphic forms for GL(2)

We begin with a presentation of the basic theory of automorphic forms for GL(2), which will be the main objects of study throughout this thesis. Most of the definitions and notation are taken from Goldfeld and Hundley [12], though there are some differences.

Definition 1.1 (Upper half plane). Let h denote the upper half plane {x + iy | x ∈ R, y > 0}.

The group SL(2, R) acts on h by Möbius transformations:

( ) a b az + b z = . c d cz + d

Definition 1.2 (Extended upper half plane). Let h∗ = h ∪ Q ∪ ∞.

10 The action of the subgroup SL(2, Z) ⊂ SL(2, R) can be extended to h∗ by defining

( ) { a b a if c ≠ 0, ∞ = c c d ∞ if c = 0,

and ( ) { a b m ∞ if cm + dn = 0, = c d n am+bn cm+dn otherwise.

For q a positive integer, we define:

Definition 1.3 (Congruence subgroup Γ0(q)).

{( ) } a b Γ (q) = ∈ SL(2, Z) | c ≡ 0 mod q . 0 c d

Definition 1.4 (Cusp). A cusp for Γ0(q) is an equivalence class in Q∪∞ with respect to the action of Γ0(q) ⊆ SL(2, Z).

Definition 1.5 (Moderate growth). Let f : h → C be a smooth function, and a ∈ Q ∪ ∞. Then f

N has moderate growth at a if, for any σa ∈ SL(2, R) such that σa∞ = a, f(σa(x + iy)) = O(y )

for some N as y → ∞.

This allows for a basic definition of automorphic functions:

Definition 1.6 (Automorphic Function). Let k ≥ 0 be an integer. An automorphic function of

weight k for Γ0(q) is a smooth function f : h → C such that: ( ) a b • If γ = ∈ Γ (q), then f(γz) = (cz + d)kf(z) for all z ∈ h; c d 0

• f has moderate growth at all cusps for Γ0(q).

11 The space of automorphic functions is equipped with the following inner product:

Definition 1.7 (Petersson inner product). Let f and g be automorphic functions of weight k for

Γ0(q). Then the Petersson inner product of f and g is

∫∫ dxdy ⟨f, g⟩ = f(z)g(z)yk , k y2 Γ0(q)\h

where the integral is taken over a fundamental domain for the action of Γ0(q) on h. The subscript k will be suppressed when it is clear from context.

While this integral will not always converge in general, we will often impose the condition

⟨f, f⟩ < ∞ on the functions studied below, ensuring that this is a true inner product in the relevant

2 setting. The Hilbert space generated by all such functions is denoted L (Γ0(q)\h, k). The study of the spectral theory of this space goes back to Selberg [30], who decomposed it as direct sum of a discrete spectrum (cusp forms), a continuous spectrum (Eisenstein series), and a residual spectrum

(residues of Eisenstein series).

The following operators act on the space of automorphic functions:

Definition 1.8 (Hecke operators). Let n be a positive integer. Then for f an automorphic function for Γ0(q), define the Hecke operator Tn by

( ) 1 ∑ ∑d az + b T f(z) = √ f . n n d ad=n b=1 (a,N)=1

Also, define ( ) −1 T− f(z) = f . 1 Nz

12 It can be verified that ∑ TmTn = T mn , d2 d|(m,n) (d,N)=1

so the Hecke operators Tn commute with each other.

For (n, N) = 1, the operators Tn preserve the space of automorphic functions and are normal operators with respect to the Petersson inner product (for details, see Goldfeld [11]). Thus, any

2 subspace of L (Γ0(q)\h, k) which is invariant under the action of the Tn for (n, N) = 1 has an

orthonormal basis consisting of simultaneous eigenfunctions of all such Tn.

1.2 Maass Forms of Weight Zero

Definition 1.9 (Non-Euclidean Laplacian). Define

( ) ∂2 ∂2 ∆ = −y2 + . ∂x2 ∂y2

The operator ∆ is invariant under the action of SL(2, R), and hence acts on the space of Γ0(q)-

invariant functions. The Maass forms are the automorphic functions which are eigenfunctions of

this operator.

Definition 1.10. A Maass form of weight zero for Γ0(q) is a smooth function f : h → C such that:

• f is an automorphic function of weight 0 for Γ0(q)in the sense of Definition 1.6,

1 2 • ∆f = ( 4 + t )f.

1 2 The notation 4 +t for the Laplace eigenvalue will be used throughout. Note that the eigenvalues

of ∆ are real and positive (Goldfeld [11], Proposition 3.3.2). Conjecturally, the t are real:

13 Conjecture 1.1 (Selberg [31]). Let f be a non-constant function on Γ0(q)\h with ∆f = λf. Then

λ ≥ 1/4.

≥ 975 ≈ At present, the best bound, due to Kim and Sarnak [18] is λ 4096 0.238 ..., so it is possible

7 that t may be purely imaginary of absolute value at most 64 .

The Fourier expansion of such a Maass form about each cusp may be defined as follows. Let { ( ) } 1 b Γ∞ =  | b ∈ Z be the stabilizer of ∞ in Γ (q). For any cusp a, we can choose 0 1 0 ∈ R ∞ −1 σa SL(2, ) such that σa = a and σa Γaσa = Γ∞, where Γa is the stabilizer of a in Γ0(q) (a

detailed construction of σa is given on p. 87 of [12]). Thus, there exists ga ∈ Γa ⊂ Γ0(q) such that ( ) 1 1 σ−1g σ = . Hence, for any Maass form f of weight zero for Γ (q), we have a a a 0 1 0

( ( ) ) 1 1 f(σ (z + 1)) = f σ z = f(g σ z) = f(σ z), a a 0 1 a a a

so the function z 7→ f(σaz) is periodic in x with period one. Thus, we may write

∑ f(σaz) = can(y)e(nx), n∈Z

where again e(θ) = e2πiθ. This facilitates the following definition:

Definition 1.11 (Maass cusp form). A Maass cusp form of weight zero for Γ0(q) is a Maass form

f of weight zero for Γ0(q) such that

• ⟨f, f⟩ < ∞,

• The Fourier coefficient ca0(y) defined above vanishes for all cusps a.

1 2 The condition ∆f = ( 4 + t )f, when applied to the Fourier expansion, shows that the Fourier

14 coefficients can(y) of a cusp form, n ≠ 0, satisfy the differential equation

( ) 2 ′′ 2 2 2 − − 2 y can(y) = 4π n y 1 t can(y).

The only solutions to this equation which are compatible with the moderate growth condition are of the form √ can(y) = a(n) yKit(2π|n|y), where K denotes the Bessel function

∫ ∞ 1 −y(t+t−1)/2 s dt Ks(y) = e t , 2 0 t and thus any cusp form f has the following Fourier-Whittaker expansion at the cusp ∞:

∑ √ f(z) = af (n) yKit(2π|n|y)e(nx). n=0̸

This is Proposition 3.5.1 in Goldfeld [11], which contains more details.

These af (n) may be used to define a Dirichlet series, but in some cases, the resulting function will not have the typical properties of an L-function, such as an Euler product. In particular, if f is a Maass cusp form of weight zero for the full modular group SL(2, Z), then the function z 7→ f(Nz) will be a Maass cusp form of weight zero for Γ0(q). However, this function has first Fourier coefficient 0, so it is impossible to normalize in such a way that the naïvely defined

L-function has an Euler product.

The remedy to this situation is the concept of newforms developed by Atkin and Lehner [1].

15 Definition 1.12. An oldform is a cusp form f such that f(z) = g(dz) for some form g on Γ0(M),

N where M is a proper divisor of N and d is a positive divisor of M .A newform is a cusp form which

is orthogonal to all oldforms with respect to the Petersson inner product.

The Hecke operators Tn (for all n, not just (n, N) = 1), preserve the space of newforms. Thus, the space of newforms has an orthonormal basis consisting of simultaneous eigenfunctions for all the Tn. Such a form has a well-behaved L-function:

Definition 1.13 (L-function). Let

∑ √ f(z) = af (n) yKit(2π|n|y)e(nx) n∈Z

be a newform of weight zero, and let

af (n) λf (n) = . af (1)

Normalizing such that λf (1) = 1 has the result that λf (n) is the eigenvalue of Tn corresponding

to f. Then the L-function associated to f is defined for ℜ(s) > 1 by

∑∞ −s L(s, f) = λf (n)n . n=1

The relation ∑ TmTn = T mn d2 d|(m,n) (d,N)=1

on the Hecke operators implies that that the Hecke eigenvalues are multiplicative in the sense that

16 λf (mn) = λf (m)λf (n) whenever (m, n) = 1. Thus, the L-function defined above has an Euler product:

Proposition 1.1 (Euler product).

∏ ( ) ∏ ( ) −s −1 −s −2s −1 L(s, f) = 1 − λf (p)p 1 − λf (p)p + p p|q p∤q ∏ ( ) ∏ ( ) ( ) −s −1 −s −1 −s −1 = 1 − λf (p)p 1 − α1,f (p)p 1 − α2,f (p)p , p|q p∤q

where the numbers αi,f (p) satisfy α1,f (p) + α2,f (p) = λf (p) and α1,f (p)α2,f (p) = 1.

The order of growth of αi,f (p), and by extension the Hecke eigenvalues λf (n), is the subject

of the Ramanujan-Petersson conjecture. This conjecture holds that |αi,f (p)| = 1, but remains an

open problem. The best result to date is the following:

Proposition 1.2 (Kim-Sarnak [18]).

7/64 |αi,f (p)| ≤ p .

Just as with the Riemann zeta function and the Dirichlet L-functions, this L-function also has

analytic continuation to the whole complex plane and satisfies a functional equation:

Proposition 1.3 (Functional equation). Let

( ) ( ) s + ϵ + it s + ϵ + it Λ(s, f) = qs/2π−sΓ Γ L(s, f), 2 2

17 1 2 where 4 + t is the Laplace eigenvalue of f, and

{ 0 if T− f = f, ϵ = 1 1 if T−1f = −f.

Then

Λ(s, f) = (−1)ϵΛ(1 − s, f).

The Generalized Riemann Hypothesis in this context asserts that all zeros of L(σ + it, f) in the critical strip 0 < σ < 1 lie on the line σ = 1/2.

In the following sections, we will need a standard basis for the space of cusp forms (all such forms, not only newforms). Using the material above, we may choose this basis as follows:

Definition 1.14 (Basis for space of cusp forms). Let {uj}j≥1 be a basis for the space of Maass cusp forms of weight zero for Γ0(q) such that:

• The basis {uj} is orthonormal with respect to the Petersson inner product;

• Each uj is an eigenfunction of all the Hecke operators Tn when (n, N) = 1;

• The basis {uj} contains a subset which is a basis for the space of newforms, and each uj

which is a newform is an eigenfunction of the Hecke operators Tn for all n;

• The Laplace eigenvalues λj (such that ∆uj = λjuj) are nondecreasing: λ1 ≤ λ2 ≤ · · · .

Further define u0 to be the constant function with Petersson norm 1. While the constant function is not a cusp form, including u0 in this basis will simplify the notation below.

18 1.3 Eisenstein Series and the Selberg Spectral Decomposition

2 In addition to the Maass cusp forms of weight zero, the space L (Γ0(q)\h) contains a continuous spectrum which may be expressed in terms of Eisenstein series. The Eisenstein series may be defined as follows:

Definition 1.15 (Eisenstein Series). Let a be a cusp for Γ0(q). For ℜ(s) > 1 and z ∈ h, let

∑ s Ea(z, s) = Im(σaγz) ,

γ∈Γa\Γ0(q)

where, as before Γa is the stabilizer of a and σa ∈ SL(2, R) is chosen such that σa∞ = a and

−1 σa Γaσa = Γ∞.

The series Ea(z, s) has the following Fourier expansion about each cusp b, as given in Kubota

[21]:

− 1 1/2 s ∑ s 1/2 Γ(s ) 1−s 2y π s− 1 2 | | 2 | | Eα(σbz, s) = δaby + π φab0(s)y + n φabn(s)Ks− 1 (2π n y)e(nx), Γ(s) Γ(s) 2 n=0̸ where ( ) ∑ ∑ nd φ (s) = c−2s e , abn c c>0 d ( ) a b the inner sum being over the congruence classes d (mod c) such that there exists ∈ c d

−1 σa Γ0(q)σb. This expansion shows that Ea(z, s) has an analytic continuation as a function of s to the entire complex plane, and that the vector whose components are the Eisenstein series Ea(z, s) for all cusps a has a functional equation for s 7→ 1 − s.

The Selberg spectral decomposition states that the Maass cusp forms, the Eisenstein series

19 1 Ea(z, 2 + ir), and the residues of Eisenstein series (in this case, just the constant function u0)

2 generate L (Γ0(q)\h) in the following way:

2 Theorem 1.1 (Selberg spectral decomposition). Let f ∈ L (Γ0(q)\h). and let {uj} be the basis of

cusp forms from Definition 1.14. Then

∫ ⟨ ⟩ ∑ 1 ∑ ∞ 1 1 f(z) = ⟨f, uj⟩uj(z) + f, Ea(·, + ir) Ea(z, + ir)dr, 4π −∞ 2 2 j≥0 a

where a varies over the set of all cusps for Γ0(q).

For more discussion of this result, see Theorem 3.16.1 in Goldfeld [11]. This immediately leads

to the following:

2 Proposition 1.4 (Parseval identity). Let f, g ∈ L (Γ0(q)\h), and let {uj} be as above. Then

∫ ⟨ ⟩ ⟨ ⟩ ∑ 1 ∑ ∞ 1 1 ⟨f, g⟩ = ⟨f, uj⟩⟨g, uj⟩ + f, Ea(·, + ir) g, Ea(·, + ir) dr. 4π −∞ 2 2 j≥0 a

20 Chapter 2

The Kuznetsov Trace Formula and Large

Sieve Inequalities for Fourier Coefficients

This chapter develops the Kuznetsov trace establishes a relationship between Fourier coefficients of Maass cusp forms and Kloosterman sums, and uses this relationship to derive the large sieve inequalities which will be used in the proof of the main theorem.

2.1 Kloosterman Sums

Definition 2.1 (Kloosterman sum). Let

( ) ∑∗ md + nd S(m, n; c) = e , c d mod c where the asterisk means that the sum is taken over residue classes such that (c, d) = 1 and d is such that dd ≡ 1 mod c.

21 These sums, used by Kloosterman in [19] to provide an improvement to the Hardy-Littlewood circle method, have a number of useful properties and applications.

Proposition 2.1. Let (c1, c2) = 1, and choose c1 and c2 such that c1c1 ≡ 1 mod c2 and c2c2 ≡

1 mod c1. Then

S(m, n; c1c2) = S(c2m, c2n; c1)S(c1m, c1n; c2).

Proof. The right hand side is

( ) ( ) ∑∗ c md + c nd ∑∗ c md + c nd e 2 1 2 1 e 1 2 1 2 c1 c2 d c d c 1 mod 1 (2 mod 2 ) ∑∗ ∑∗ c c md + c c nd + c c md + c c nd = e 2 2 1 2 2 1 1 1 2 1 1 2 . c1c2 d1 mod c1 d2 mod c2

The numerator is congruent to md1+nd1 mod c1 and md2+nd2 mod c2. Thus, for each d mod c1c2 such that (d, c1c2) = 1, if d1 and d2 denote the residue classes of d mod c1 and mod c2, respectively, then the numerator will be congruent to md + nd mod c1c2 by the Chinese Remainder Theorem.

Since, again by the Chinese Remainder Theorem, the d mod c1c2 such that (d, c1c2) = 1 are in bijective correspondence with the resulting pairs (d1, d2) appearing in the above sum, it follows that the sum is just S(m, n; c1c2).

This multiplicativity relation reduces the problem of estimating Kloosterman sums in general to that of estimating Kloosterman sums where c is a prime power. In this case, the work of Weil

[39] allows the sums to be bounded by means of the Riemann hypothesis for function fields. The result is:

22 Proposition 2.2 (Weil bound for Kloosterman sums).

|S(m, n; c)| ≤ τ(c)(m, n, c)1/2c1/2,

where τ(c) is the number of positive divisors of c and (m, n, c) is the greatest common divisor of

m, n, and c.

The following useful identity may be proved (as in Kuznetsov [22], Theorem 4) as an applica-

tion of the Kuznetsov trace formula and the identity

∑ TmTn = T mn d2 d|(m,n)

for the Hecke operators for SL(2, Z)\h. An elementary proof is also possible, as shown by Matthes

[24].

Proposition 2.3. ( ) ∑ mn c S(m, n; c) = d · S 1, , . d2 d d|(m,n,c)

The Kloosterman sums may be used to define the Kloosterman zeta function

∑∞ −2s Zm,n(s) = S(m, n; c)c . c=1

The existence of an analytic continuation for the Kloosterman zeta function relates to the ques- tion of Conjecture 1.1 on the eigenvalues of the Laplace operator. Selberg [31] showed that, for real σ > 1/2, the Kloosterman zeta function Zm,n(s) has analytic continuation to ℜ(s) > σ if and

23 only if the least positive eigenvalue of ∆ is at least σ(1 − σ). The bound of Proposition 2.2 may be used to show continuation to ℜ(s) > 3/4, which gives a lower bound of 3/16 for the eigenvalues.

The application of Kloosterman sums to large sieve inequalities for Fourier coefficients requires the development of large sieve type inequalities for bilinear forms involving Kloosterman sums.

The technique for this is due to Iwaniec [16], with some modifications for the purposes of this thesis.

The proof will require the use of the analytic large sieve. Recall the large sieve of Montgomery and Vaughan mentioned in the introduction:

Theorem 2.1 (Montgomery-Vaughan [26]). Let α1, . . . , αR be a set of real numbers such that

||αr − αs|| ≥ δ

for r ≠ s, where || · || denotes the distance to the nearest integer, and let

M∑+N S(α) = ane(nα). n=M+1

Then ∑R 1 M∑+N |S(α )|2 ≤ (N + ) |a |2. r δ n r=1 n=M+1

Taking αr to be the fractions r/c for all r mod c then yields, using δ = 1/c in the above:

Proposition 2.4. M+N ( ) 2 M+N ∑ ∑ r ∑ a e n ≤ (N + c) |a |2. n c n r mod c n=M+1 n=M+1

This may be used to prove the following inequality for Kloosterman sums:

24 Proposition 2.5. Let

( √ ) M∑+N M∑+N 2 mn B(θ, c, M, N) = b b S(m, n, c)e θ , m n c m=M+1 n=M+1

where N ≤ M are positive integers and θ > 0 is a real number. Suppose c > M 1−ϵ. Then

M∑+N ϵ 2 |B(θ, c, M, N)| ≪ M (c + N) |bn| . n=M+1

√ N ≤ ≤ mn ≤ Proof. Let α = M 1. Then 1 M 1+α for all m, n appearing in the sum. Thus, attaching √ mn a smooth weight function η( M ) such that

{ 1 if 1 ≤ x ≤ 1 + α, η(x) = ≤ − α ≥ 0 if x 1 2 or x 1 + 2α. will not affect the sum. By the Mellin inversion formula,

∫ 1 1+i∞ η(x)e(2θc−1Mx) = R(s)x−sds, 2πi 1−i∞

where ∫ ∞ R(s) = η(x)e(2θc−1Mx)xs−1dx. 0

For s = 1 + it, this is bounded for all t and by t−2 for |t| > 16πθc−1M. Applying this Mellin

25 inversion to the sum gives

(√ ) ( √ ) M∑+N M∑+N mn 2 mn B(θ, c, M, N) = b b S(m, n, c)η e m n M c m=M+1 n=M+1 ∫ (√ )− 1 1+i∞ M∑+N M∑+N mn s = bmbnS(m, n, c)R(s) ds 2πi 1−i∞ M m=M+1 n=M+1 ∫ M+N ( ) M+N ( ) 1+i∞ ∑∗ ∑ ∑ −s/2 md −s/2 nd s ≪ bmm e bnn e M R(s)ds. − ∞ c c 1 i d mod c m=M+1 n=M+1

By Proposition 2.4,

M+N ( ) M+N ( ) M+N ∑∗ ∑ md ∑ nd ∑ b m−s/2e b n−s/2e ≤ M −1(N + c) |b |2. m c n c n d mod c m=M+1 n=M+1 n=M+1

Thus ∫ 1+i∞ M∑+N 2 |B(θ, c, M, N)| ≪ R(s)ds (N + c) |bn| , − ∞ 1 i n=M+1 and the result follows from the bounds on R and the assumption c > M 1−ϵ.

In the case of smaller c, we have the following:

Proposition 2.6. Let B(θ, c, M, N) be as in Proposition 2.5, where again N ≤ M, and now assume c ≤ M 1−ϵ. Assume further that 0 < θ < 2. Then

M∑+N − 1 1 1 1 +ϵ 2 |B(θ, c, M, N)| ≪ θ 2 c 2 N 4 M 4 |bn| . n=M+1

26 Proof. By Proposition 2.3,

( ) ∑ c M N B(θ, c, M, N) = d · X θ, , , , d d d d|c

where for xn = bdn, q = c/d, K = M/d, L = N/d,

( √ ) ∑ 2 mn X (θ, q, K, L) = x x S(1, mn; q)e . m n q K

By Cauchy-Schwartz,

( ) ( ) ( √ ) 2 ∑ ∑ n ∑ 2 mn |X(θ, q, K, L)|2 ≤ |x |2 η x S(1, mn; q)e n K m q K

( ) ( √ √ ) n h m − h m 2( m − m )θ√ f(n) = η e 1 1 2 2 n + 1 2 n . K q q

( ) √ n For brevity, define A and B so that f(n) = η K e (An + B n). By the Poisson summation

formula, ∑ ∑ f(n) = fb(u), n∈Z u∈Z

27 where ∫ ( ) ( ) ∞ t √ fb(u) = η e (A − u)t + B t dt. −∞ K ( ) ̸ | − | ≥ 1 t ̸ Suppose u = A. Then A u q . Also, for all t such that η K = 0 (and thus t > K/2), √ √ | | B√ < θ K√+L− K ≪ 1 Integrating by parts p times gives: 2 t q K/2 q

  ( ) ∫ (( ( ) )′ )′ ′ − p ∞ b 1 ··· t 1 1 ···  f(u) = η B B 2πi −∞ K A − u + √ A − u + √ 2 t 2 t ( √ ) · e (A − u)t + B t dt,

so

fb(u) ≪ K(|A − u|K)−p.

Thus ( )− ( )− ∑ K p M M p M fb(u) ≪ K = ≪ M −ϵp ≪ K−1 q d c d u≠ A for p sufficiently large. The remaining case is u = A, which from the definition of A implies

h1m1 ≡ h2m2 mod q. In this case,

( ) ∑∗ h − h e 1 2 ≪ (m − m , q), q 1 2 h1,h2 mod q

where the sum is taken over residue classes with h1m1 ≡ h2m2 mod q. b If m1 = m2 then trivially f(A) ≪ L. Otherwise, integration by parts gives

∫ ( ) ( ) √ √ ∞ t √ L q KL fb(A) = η e B t dt ≪ ≪ . −∞ K |B| θ|m1 − m2|

28 Combining everything above gives

( ) ∑ ∑∗ h − h ∑ x x e 1 2 f(n) m1 m2 q K

and thus ( ) √ ∑ 2 2 −1 ϵ 2 |X(θ, q, K, L)| ≪ θ qK KL |xn| . K

Taking square roots and summing over d|c gives the result.

2.2 Kuznetsov Trace Formula

To relate Kloosterman sums to Fourier coefficients of cusp forms, we use the trace formula devel-

oped by Kuznetsov [22]. The key to this formula is the non-holomorphic Poincaré series, which is

defined as follows (see Selberg [31]):

Definition 2.2. The non-holomorphic Poincaré series (at the cusp ∞) for Γ0(q), for integer m ≥ 0,

is ∑ s Pm(z, s) = Im(γz) e(mγz).

γ∈Γ∞\Γ0(q)

This is a generalization of the Eisenstein series, which results from setting m = 0. The Poincaré series converges absolutely for ℜ(s) > 1 and satisfies Pm(γz, s) = Pm(z, s) by construction.

2 Furthermore, it is square integrable, so Pm(·, s) ∈ L (Γ0(q)\h).

29 The Kuznetsov trace formula is derived by computing the inner product ⟨Pm(·, s1),Pn(·, s2)⟩ in two ways. First, it may be computed directly by means of the Fourier expansion of Pm(z, s),

which exists because Pm(z, s) is automorphic. The expansion is as follows (details can be found

in Kuznetsov [22]), and accounts for the appearance of Kloosterman sums in the trace formula:

Proposition 2.7. Suppose ℜ(s) > 1 and m ≥ 0. Then

∑ Pm(z, s) = Bn(m, y, s)e(nx), n∈Z

where

Bn(m, y, s) = ∫ ( ) ∑ ∞ m s −2πny s −2s 2 2 −s − − δmny e + y c S(m, n; c) (x + y ) e nx 2 dx. −∞ c (x + iy) c>0 c≡0 mod q

Based on the fact that, by the Weil bound (Proposition 2.2), the Kloosterman zeta function ∑ ∞ −2s ℜ Zm,n(s) = c=1 S(m, n; c)c converges for (s) > 3/4, this formula may be used to give an

analytic continuation of the Poincaré series Pm(z, s) to ℜ(s) > 3/4 as well. This expansion gives

the following formula for the inner product of two Poincaré series:

Proposition 2.8. Let m, n ≥ 1. Then

∫ ∞ s2−2 −2πny ⟨Pm(·, s1),Pn(·, s2)⟩ = Bn(m, y, s1)y e dy. 0

30 Proof. Let D be a fundamental domain for the action of Γ0(q) on h. Then

∫∫ ∑ dxdy ⟨P (·, s ),P (·, s ⟩ = P (z, s ) Im(γz)s2 e(nz) m 1 n 2 m 1 y2 D γ∈Γ∞\Γ0(q) ∑ ∫∫ − dxdy = P (γ 1z, s )ys2 e(nz) m 1 y2 ∈ \ γ Γ∞ Γ0(q) γD ∫∫

s2−2 −2πny = Pm(z, s1)e(−nx)y e dxdy, E

where E is a fundamental domain for the action of Γ∞. Such a fundamental domain is given by the vertical strip 0 ≤ x < 1, y > 0 Replacing Pm(z, s1) with its Fourier expansion then gives

∫ ∫ ∞ 1 ∑ s2−2 −2πmy ⟨Pm(·, s1),Pn(·, s2)⟩ = Bk(m, y, s1)e(kx)e(−nx)y e dxdy. 0 0 k∈Z

The integral over x isolates the case k = n, finishing the proof.

Substituting the expression from Proposition 2.7 for Bk(m, y, s1) allows this inner product to be expressed in terms of Kloosterman sums. Again using the Weil bound for Kloosterman sums, this allows for the analytic continuation of ⟨Pm(·, s1),Pn(·, s2)⟩ for ℜ(s1) > 3/4, ℜ(s2) > 3/4, and, hence, it makes sense to consider s1 = 1+it, s2 = 1−it. In this case, the explicit calculations were carried out by Kuznetsov [22], resulting in the following:

Proposition 2.9. Let m, n ≥ 1. Then

( ) ∫ ( √ ) it ∑ i ⟨ · · ⟩ δmn − n S(m, n; c) 4π mn dv Pm( , 1 + it),Pn( , 1 + it) = 2i 2 K2it v . 4πn m c − c v c>0 i c≡0 mod q

The inner product ⟨Pm(·, s1),Pn(·, s2)⟩ may also be computed using the Selberg spectral

31 2 decomposition (Theorem 1.1), taking advantage of the fact that Pm(·, s) ∈ L (Γ0(q)\h). This

version of the computation can be expressed in terms of the Fourier coefficients of cusp forms,

since the inner product of a Poincaré series Pm and a cusp form relates to the mth Fourier coeffi-

cent as follows:

∑ √ Proposition 2.10. Let f(z) = af (n) yKit(2π|n|y)e(nx) be a Maass cusp form of weight zero n=0̸ 1 2 ≥ ℜ for Γ0(q) with Laplace eigenvalue 4 + t . Then, for m 1 and (s) > 1,

− 1 − 1 − 1 −s 1 Γ(s 2 + it)Γ(s 2 it) ⟨P (·, s), f⟩ = (4πm) 2 π 2 a (m) . m f Γ(s)

Proof. The proof is similar to the proof of Proposition 2.8, and the same notation will be used.

∫∫ ∑ dxdy ⟨P (·, s), f⟩ = Im(γz)se(mγz)f(z) m y2 D γ∈Γ∞\Γ0(q) ∫∫ ∑ dxdy = Im(z)se(mz)f(γ−1z) y2 ∈ \ γ Γ∞ Γ0(q) γD ∫∫ = f(z)e(mx)ys−2e−2πmydxdy

∫E ∫ ∞ 1 ∑ s− 3 −2πmy = af (n)Kit(2π|n|y)e(−nx)e(mx)y 2 e dxdy 0 0 n=0̸ ∫ ∞ s− 3 −2πmy = af (m) Kit(2πmy)y 2 e dy. 0

The computation

∫ ∞ − 1 − 1 − s− 3 −y 1 1 −s Γ(s 2 + it)Γ(s 2 it) Kit(y)y 2 e dy = π 2 2 2 0 Γ(s)

32 (Kubota [21], p. 50) finishes the proof.

A similar argument applies for the Eisenstein series, giving:

Proposition 2.11. Let m ≥ 1 and ℜ(s) > 1, and let a be a cusp for Γ0(q). Then

⟨ ( ) ⟩ 1 P (·, s),E ·, + ir m a 2 ( ) − 1 − 1 − 2−2s 1 −s−ir 3 −s−ir 1 Γ(s 2 + ir)Γ(s 2 ir) = 2 m 2 π 2 φ ∞ + ir . a m 1 − 2 Γ(s)Γ( 2 ir)

The Parseval identity (Proposition 1.4) and the above computations immediately give:

∑ √ | | Proposition 2.12. Let uj(z) = n=0̸ aj(n) yKit(2π n y)e(nx) be as in Definition 1.14, where

1 2 ≥ the Laplace eigenvalue of uj is 4 + tj , and let m, n 1. Then

( 1 1 −s1 −s2 ∑ ⟨ · · ⟩ πm 2 n 2 Pm( , s1),Pn( , s2) = s +s −1 aj(m)aj(n)Λ(s1, s2; tj) (4π) 1 2 Γ(s1)Γ(s2) j≥1 ) ∫ ∞ ( ) ( ) ( ) ∑ ir n 1 1 Λ(s1, s2; r) + φ ∞ + ir φ ∞ + ir dr , a m a n 1 2 −∞ m 2 2 |Γ( + ir)| a 2 where

( ) ( ) ( ) ( ) 1 1 1 1 Λ(s , s ; r) = Γ s − + ir Γ s − − ir Γ s − + ir Γ s − − ir . 1 2 1 2 1 2 2 2 2 2

Specializing as above to the case s1 = 1 + it, s2 = 1 − it gives:

33 Proposition 2.13. Keeping the notation as above,

( ( ) ∑ 1 n it sinh πt a (m)a (n) ⟨P (·, 1 + it),P (·, 1 + it)⟩ = √ π j j m n 4 mn m t cosh π(t − t) cosh π(t + t) j≥1 j j ∫ ( ) ( ) ) ∑ ∞ ( ) 1 1 n ir φ ∞m + ir φ ∞n + ir + a 2 a 2 cosh πrdr . −∞ m cosh π(r − t) cosh π(r + t) a

The following version of the Kuznetsov trace formula (Deshouillers-Iwaniec [6], Lemma 4.7) then follows by equating the expressions in Propositions 2.9 and 2.13 and dividing both sides by ( ) √1 n it sinh πt 4 mn m t :

Theorem 2.2 (Kuznetsov trace formula). Let {uj} be a basis as in Definition 1.14 for the space of

1 2 Maass cusp forms of weight zero for Γ0(q), with Laplace eigenvalues 4 +tj and Fourier expansions ∑ √ | | ≥ ∈ R uj(z) = n=0̸ aj(n) yKit(2π n y)e(nx). Let m, n 1 be integers, and let t . Then

√ ∫i ( √ ) −2it ∑ 4π mn 4π mn dv t S(m, n; c) K v + δ sinh πt c2 2it c v mn π sinh πt c>0 − c≡0 mod q i ∫∞ ( ) ( ) ∑ ∑ ( ) 1 1 a (m)a (n) n ir φ ∞m + ir φ ∞n + ir = π j j + a 2 a 2 cosh πrdr. cosh π(tj − t) cosh π(tj + t) m cosh π(r − t) cosh π(r + t) j≥1 a −∞

Here as usual a varies over the cusps of Γ0(q).

2.3 Large Sieve Inequality for Maass Forms

Theorem 2.2 allows for sums involving Fourier coefficients of the Maass cusp forms {uj} of weight zero (as in Definition 1.14) to be studied by means of the Kloosterman sums appearing on the opposite side of the trace formula. The result is the following large sieve inequality:

34 Theorem 2.3. With the notation from Theorem 2.2, let {bn} be any sequence of complex numbers, and let M ≥ N be integers. Then

∑ M∑+N 2 M∑+N 1 2 ϵ 2 bnaj(n) ≪ (K + M N) |bn| . cosh πtj tj ≤K n=M+1 n=M+1

−(t/K)2 Proof. Multiplying both sides of Theorem 2.2 by t sinh πte bmbn, and integrating over t from

−∞ to ∞, we have ([6], (5.5)):

∑ M∑+N 2 tj + 1 − 2 (tj /K) e bnaj(n) ≥ cosh πtj j 1 n=M+1 ∫ ( ) 2 ∑ ∞ M∑+N −(r/K)2 ir 1 + (|r| + 1)e bnn φa∞n + ir dr −∞ 2 a n=M+1 √ ( √ ) M∑+N ∑ 1 M∑+N M∑+N 4π mn 4π mn ≪ K3 |b |2 + b b S(m, n; c)Φ , n c m n c c n=M+1 c>0 m=M+1 n=M+1 c≡0 mod q where ∫ ∫ ∞ i 2 −(t/K)2 dv Φ(x) = t e K2it(xv) . −∞ −i v

The function Φ has the following representations ([6], (5.6), (5.7)):

∫ ∞ √ 2 Φ(x) = πiK3 e−(ξK) ξ tanh ξ sin(x cosh ξ)dξ ≪ 1 ∫0 √ ∞ ( ) 2 dξ xΦ(x) = πiK3 e−(ξK) 1 − ξ tanh ξ − 2ξ2K2 cos(x cosh ξ) ≪ xK−2. 0 cosh ξ

The first representation will be used for c > K−2M and the second for c ≤ K−2M. Let

√ ( √ ) M∑+N M∑+N 4π mn 4π mn ϕ(c, K, M, N) = b b S(m, n; c)Φ . m n c c m=M+1 n=M+1

35 For c > M 2, the Weil bound (Proposition 2.2) gives

M∑+N − 1 +ϵ 2 2 Φ(c, K, M, N) ≪ c 2 M |bn| . n=M+1

For M 1−ϵ < c ≤ M 2, Proposition 2.5 gives

M∑+N ϵ 2 Φ(c, K, M, N) ≪ M N |bn| . n=M+1

For K−2M < c ≤ M 1−ϵ, the first integral representation above, combined with Propositions 2.5

(for ξ ≤ 1) and 2.6 (for ξ ≥ 1), gives

M∑+N ϵ −K2 −1 2 1 5 1 2 Φ(c, K, M, N) ≪ M (Ke c MN + K c 2 M 4 N 4 ) |bn| . n=M+1

Finally, for c ≪ K−2M, the second integral above gives

M∑+N ϵ −K2 −1 1 1 1 2 Φ(c, K, M, N) ≪ M (Ke c MN + c 2 M 4 N 4 ) |bn| . n=M+1

Summing all the above estimates gives

∑ M∑+N 1 2 |ϕ(c, K, M, N)| ≪ KN(M ϵ + MKe−K ) |b |2 , c n c>0 n=M+1 c≡0 mod q and thus, substituting into the trace formula expression above and discarding the Eisenstein series

36 term gives

∑ M∑+N 2 M∑+N tj + 1 2 ϵ −K2 2 bnaj(n) ≪ K(K + M N + MNKe ) |bn| . cosh πtj tj ≤K n=M+1 n=M+1

Since the left-hand side is nondecreasing as a function of K, we may replace K by K + M ϵ on the right hand side, reducing the expression to

∑ M∑+N 2 M∑+N tj + 1 2 ϵ 2 bnaj(n) ≪ K(K + M N) |bn| , cosh πtj tj ≤K n=M+1 n=M+1 and then by partial summation

∑ M∑+N 2 M∑+N 1 2 ϵ 2 bnaj(n) ≪ (K + M N) |bn| . cosh πtj tj ≤K n=M+1 n=M+1

37 Chapter 3

Zero Density Estimate for L-functions

The aim of this chapter is to apply the large sieve inequality in Theorem 2.3 to derive a zero density estimate for the corresponding L-functions. The technique is derived from Gallagher’s proof in [10] of the analogous estimate given in Theorem 0.4 for Dirichlet L-functions. The proof combines a bound derived from the large sieve inequality with the zero-counting method developed by Turán

[33, 35, 34].

3.1 Turán’s Method

The following theorem will be used to link the zeros of L-functions to sums over Fourier coeffi- cients. It is essentially Theorem 5 in Gallagher [10] adapted from the case of Dirichlet L-functions to the case of L-functions for GL(2) automorphic forms. A few facts will be needed which carry over directly from the theory of Dirichlet L-functions, beginning with the following formula for the logarithmic derivative (as in Prachar [29], Satz 4.1, p. 225), which follows from the functional equation (Proposition 1.3):

38 Proposition 3.1. Let L(s, f) be the L-function as in Definition 1.13 of a newform for Γ0(q), and let w = 1 + iv. Then L′ ∑ 1 (s, f) = + O(log q(|v| + 2)), L s − ρ ρ where the sum is over zeros ρ of L(s, f) in the disk |ρ − w| ≤ 1.

This leads to the following bound on the density of zeros in small disks near the critical line, which is due to Linnik in the case of Dirichlet L-functions (Prachar [29], p. 331, Lemma 2.1, see also Kowalski and Michel [20], Lemma 21):

Proposition 3.2. Let L(s, f) be as above, and let Q(f, v, r) be the number of zeros of L(s, f) in the disc |s − (1 + iv)| ≤ r. Then

Q(f, v, r) ≪ r log q(|v| + 2) + 1.

The following theorem on power sums is due to Turán [35]:

Proposition 3.3. Let m ≥ 0 be an integer, {bj} be an arbitrary sequence of complex numbers, and

|z1| ≥ |z2| ≥ · · · ≥ |zn|.

Then there exists an integer ν with m + 1 ≤ ν ≤ m + n and

( ) n n | ν ν ··· ν| ≥ | |ν | ··· | b1z1 + b2z2 + + bnzn z1 min b1 + + bj . 24e2(m + 2n) 1≤j≤n

The main theorem of this section is as follows:

39 Theorem 3.1. Let L(s, f) be as above, and let w = 1 + iv with |v| ≤ T . Then there exist positive

| − | ≤ 1 ≤ ≤ constants r0, A, B, C such that if L(s, f) has a zero in the disc s w r, where log T r r0,

≥ A then for all x T , ∫ B x ∑ a (p) log p dy f ≫ x−Cr log2 x, a (1)pw y x x

where the sum is taken over primes p.

Proof. This proof follows Gallagher [10], with slight modifications for the present setting. By

Proposition 3.1, for |s − w| ≤ 1/2,

L′ ∑ 1 (s, f) = + O(log T ). L s − ρ |ρ−w|≤1

Then, by Cauchy’s inequality for Taylor series coefficients, for |s − w| ≤ 1/4

1 dk L′ ∑ 1 (s, f) = (−1)k + O(4k log T ). k! dsk L (s − ρ)k+1 |ρ−w|≤1

Take s = w+r in this formula (as we may assume r < 1/4). Let λ be such that r ≤ λ ≤ 1/4. Then

by Proposition 3.2, the number of ρ in the above sum with 2jλ < |ρ − w| ≤ 2j+1λ is ≪ 2jλ log T ,

1 ≪ j −(k+1) | − | and for such ρ, (s−ρ)k+1 (2 λ) . Thus the contribution of all terms with ρ w > λ is

∑∞ ≪ (2jλ)−k log T ≪ λ−k log T. j=0

Thus, since λ ≤ 1/4, we may eliminate the terms with |ρ − w| > λ from the sum and write

1 dk L′ ∑ 1 (s, f) = (−1)k + O(λ−k log T ). k! dsk L (s − ρ)k+1 |ρ−w|≤λ

40 Again using Proposition 3.2, the number of remaining zeros is ≤ A1λ log T for some constant A1.

If there is a zero in the disc |s − w| ≤ r, then minρ |s − ρ| ≤ 2r. Now apply proposition 3.3, with

1 | | ≥ 1 ≥ the zj equal to s−ρ , the bj = 1, and z1 2r . Taking m A1λ log T , this implies that there exists

k with m ≤ k ≤ 2m and

∑ 1 −(k+1) ≥ (Dr) (s − ρ)k+1 |ρ−w|≤λ

for positive constant D. Choosing λ = A2Dr (setting r0 as necessary to assure that λ ≤ 1/4) for

−k sufficiently large A2, the above sum dominates the remainder term O(λ log T ) (using the fact

≥ 1 that r log T ), and thus 1 dk L′ (s, f) ≫ (Dr)−(k+1), k! dsk L

where still m ≤ k ≤ 2m and m ≥ A1λ log T = Er log T where E = A1A2D.

Using the Euler product in Proposition 1.1, we have

∞ L′ ∑ (s, f) = Λ (n)n−s, L f n=1

where { λ (p)j log p if p|q, Λ (pj) = f f j j (αf,1(p) + αf,2(p) ) log p if p ∤ q

j 7 j+ϵ and Λf (n) = 0 for n not a prime power. In particular, Λf (p) = λf (p) log p and Λ(f)(p ) ≪ p 64

by Proposition 1.2. Thus the above bound may be written as

∞ ∑ Λ (n) D−k f p (r log n) ≫ , nw k r n=1

41 where e−uuk p (u) = . k k!

Using Stirling’s formula, it can be shown that there exist constants B1, B2, such that the function pk

−k −k −u/2 satisfies pk(u) ≤ (2D) for u ≤ B1k and pk(u) ≤ (2D) e for u ≥ B1k. Set A = B1E. For

≥ A −1 ≥ x T , choose m = B1 r log x, so that m Er log T as required and there exists an appropriate

k with m ≤ k ≤ 2m. Let B = 2B2/B1. Then it follows that:

∑ Λ (n) ∑ Λ (n) k f p (r log n) ≪ (2D)−k f ≪ (2D)−k , nw k n r n≤x n≤x ∑ Λ (n) ∑ Λ (n) (2D)−k f ≪ −k f ≪ w pk(r log n) (2D) 1+ r . n n 2 r n>xB n>xB

The second step in each line above requires the prime number theorem for the function L(s, f) in ∑ ≪ the form n≤x Λf (n) x, which follows in particular from the zero-free region established by

Hoffstein and Ramakrishnan [15]. Thus the terms for n ≤ x and n > xB may be discarded, leaving

∑ Λ (n) D−k f p (r log n) ≫ . nw k r x

j 7 j+ϵ j Finally, the bound Λ(f)(p ) ≪ p 64 means the terms for p with j ≥ 2 may be safely discarded

as well. Set ∑ Λ (p) ∑ λ (p) log p ∑ a (p) log p S(y) = f = f = f . pw pw a (1)pw x

Then the left-hand side of the above inequality is

∫ B ∫ B x x dy B B − ′ pk(r log y)dS(y) = pk(r log x )S(x ) S(y)pk(r log y)r . x x y

42 ≪ (2D)−kk ′ ≪ As before, the first term on the right is r , and then the bound pk 1 gives

∫ xB −k dy D 2 | | ≫ ≫ −Cr log x S(y) 2 x . x y r

3.2 Exponential Sums with Fourier Coefficients

The next step in the proof of the main theorem is a bound on certain exponential sums involving

the Fourier coefficients of cusp forms (this is an analogue of Gallagher’s Theorem 2 in [10]).

The bound will use the following theorem from Gallagher [10] on exponential sums:

∑∞ it Theorem 3.2 (Gallagher [10], Theorem 1). Let S(t) = ann be an absolutely convergent n=1 Dirichlet series. Then

∫ ∫ 2 T ∞ ∑ 2 2 dy |S(t)| dt ≪ T an . −T 0 y y

Applying this to sums over Fourier coefficients gives the following:

Theorem 3.3. Let {uj} be a basis as in Definition 1.14 for the space of Maass cusp forms of

1 2 weight zero for Γ0(q), such that uj has Laplace eigenvalue 4 + tj and Fourier expansion uj(z) = ∑ √ ∑∞ | | { } n=0̸ aj(n) yKit(2π n y)e(nx). Let bn be any sequence such that the series bnaj(n) con- n=1 verges absolutely. Then

∫T ∞ 2 ∞ ∑ 1 ∑ ∑ b a (n)nit dt ≪ (K2T + n1+ϵ) |b |2 . πt n j n | |≤ cosh j tj K −T n=1 n=1

43 Proof. By Theorem 3.2, the left-hand side is

2 ∫ 1/T ∞ ∑ ye∑ 2 1 dy ≪ T bnaj(n) . (3.1) 0 cosh πtj y |tj |≤K y

Applying Theorem 2.3 then gives

2 ∑ ye∑1/T ( ) ye∑1/T 1 2 ϵ 1/T 2 bnaj(n) ≪ K + y (e − 1) |bn| . cosh πtj |tj |≤K y y

2 Substituting this in (3.1), we see that the coefficient of |bn| is

∫ ∫ n dy ( ) n ≪ T 2K2 + T 2 e1/T − 1 yϵdy ne−1/T y ne−1/T

( )( )1+ϵ ≪ K2T + T 2 e1/T − 1 1 − e−1/T n1+ϵ

≪ K2T + n1+ϵ

for T sufficiently large.

3.3 Proof of the Main Theorem

The results of the two previous sections may now be combined just as in Gallagher [10] to prove

Theorem 0.6.

1 ≤ − ≤ − ≪ 1 Proof. It suffices to consider the case where log T 2(1 α) r0, since for 1 α log T , the left-hand side is a decreasing function of α while the right-hand side is essentially constant. Thus

max(A,3) we may take r = 2(1 − α) in Theorem 3.1. Choose x = T . Then if L(s, fj) has a zero in

44 the disc |s − w| ≤ r, we have

∫ B x ∑ a (p) log p dy xCr log−2 x j ≫ 1, a (1)pw y x x≤p≤y j where the sum is over primes p. Applying Cauchy-Schwarz, and letting c = 4C max(A, 3), we obtain ∫ B 2 x ∑ a (p) log p dy T c(1−α) log−3 T j ≫ 1. a (1)pw y x x≤p≤y j

Each disc |s − w| ≤ 2(1 − α) contains ≪ ((1 − α) log T ) zeros, and each zero with real part at least 1 − α will be contained in the disc as w ranges over an interval of length ≫ 1 − α along the line w = 1 + iv. Thus we obtain

∫ B ∫ 2 x T ∑ a (p) log p dy ≪ c(1−α) −2 j Nj(α, T ) T log T 1+iv dv . − a (1)p y x T x≤p≤y j

The basis {fj} may be assumed to be a subset of a basis {uj} as in Definition 1.14. Thus,

∫ B ∫ 2 ∑ |a (1)|2 ∑∗ 1 x T ∑ a (p) log p dy j N (α, T ) ≪ T c(1−α) log−2 T j dv , j 1+iv cosh πtj cosh πtj x −T ≤ ≤ p y |tj |≤T |tj |≤T x p y

where the sum on the left hand side is over the basis {fj} of newforms and the sum on the right hand side is over the basis {uj} of all cusp forms. Now, applying Theorem 3.3 to the inner integral

log p with bp = p and bn = 0 otherwise, we obtain

∫ B ( ) ∑∗ | |2 x ∑ 2 aj(1) c(1−α) −2 3 1+ϵ log p dy Nj(α, T ) ≪ T log T (T + p ) . cosh πtj x ≤ ≤ p y |tj |≤T x p y

45 Recalling that x ≥ T 3, we see that the integral is

∑ ≪ log T pϵ log2 p ≪ T ϵ, x≤p≤xB giving the theorem.

46 Bibliography

[1] A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160.

[2] V. Blomer, Applications of the Kuznetsov formula on GL(3), Invent. Math. 194 (2013), 673– 729.

[3] E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225.

[4] R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), 1–18.

[5] H. Davenport and H. Halberstam, The values of a trigonometrical polynomial at well spaced points, Mathematika 13 (1966), 91–96.

[6] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219–288.

[7] W. Duke and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139 (2000), no. 1, 1–39.

[8] P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Math- ematica, vol. IV (INDAM, Rome, 1968/69), Academic Press, 1970, pp. 59–72.

[9] P. X. Gallagher, Bombieri’s mean value theorem, Mathematika 15 (1968), 1–6.

[10] , A large sieve density estimate near σ = 1, Invent. Math. 11 (1970), 329–339.

[11] D. Goldfeld, Automorphic forms and L-Functions for the group GL(n, R), Cambridge Uni- versity Press, 2006.

[12] D. Goldfeld and J. Hundley, Automorphic representations and L-functions for the general linear group, vol. 1, Cambridge University Press, 2011.

47 [13] D. Goldfeld and A. Kontorovich, On the GL(3) Kuznetsov formula with applications to sym- metry types of families of L-functions, Automorphic Representations and L-Functions, Tata Institute, 2013, pp. 263–310.

[14] D. A. Goldston, J. Pintz, and C. Y. Yildirim, Primes in tuples I, Ann. of Math. 170 (2009), 819–862.

[15] J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms, Int. Math. Res. Not. 6 (1995), 279–308.

[16] H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function, Sém. Th. Nb. Bordeaux, no. 18, 1979-1980.

[17] M. Jutila, A statistical density theorem for L-functions with applications, Acta Arith. 16 (1969), 207–216.

[18] H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Am. Math. Soc. 16 (2003), no. 1, 139–183.

[19] H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Math. 49 (1926), 407–464.

[20] E. Kowalski and P. Michel, Zeros of families of automorphic L-functions close to 1, Pac. J. Math. 207 (2002), no. 2, 411–431.

[21] T. Kubota, Elementary theory of Eisenstein series, Kodansha, 1973.

[22] N. V.Kuznetsov, Petersson’sconjecture for cusp forms of weight zero and Linnik’sconjecture. Sums of Kloosterman sums, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334–383.

[23] Yu. V. Linnik, The large sieve, Dokl. Akad. Nauk SSSR 30 (1941), 292–294.

[24] R. Matthes, An elementary proof of a formula of Kuznecov for Kloosterman sums, Results. Math. 18 (1990), 120–124.

[25] J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015), 383–413.

[26] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134.

[27] , The exceptional set in Goldbach’s problem, Acta Arith. 27 (1975), 353–370.

48 [28] R. J. Lemke Oliver and J. Thorner, Effective log-free zero density estimates for automorphic L-functions and the Sato-Tate conjecture, preprint, 2016.

[29] K. Prachar, Primzahlverteilung, Springer-Verlag, 1957.

[30] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

[31] , On the estimation of Fourier coefficients of modular forms, Proc. Sympos. in Pure Math., vol. VIII, Amer. Math. Soc., 1965, pp. 1–15.

[32] H. Tang, Zero density of L-functions related to maass forms, Front. Math. China 8 (2013), no. 4, 923–932.

[33] P. Turán, Eine neue Methode in der Analysis und deren Anwendungen, Akadémiai Kiadó, 1953.

[34] , On a density theorem of Yu. V. Linnik, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 165–179.

[35] P. Turán and V. T. Sós, On some new theorems in the theory of Diophantine approximations, Acta Math. Hung. 6 (1955), 241–253.

[36] R. C. Vaughan, An elementary method in prime number theory, Acta Arith. 37 (1980), 111– 115.

[37] A. I. Vinogradov, The density hypothesis for Dirichlet L-series, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903–934.

[38] , Correction to the paper of A. I. Vinogradov “On the density hypothesis for Dirichlet L-series”, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 719–720.

[39] A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. 34 (1948), 204–207.

[40] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443– 551.

[41] M. P. Young, Bilinear forms with GL3 Kloosterman sums and the spectral large sieve, Int. Math. Res. Not. 21 (2016), 6453–6492.

[42] Y. Zhang, Bounded gaps between primes, Ann. of Math. 179 (2014), 1121–1174.

49