the l4 norm of the eisenstein series

The L4 Norm of the Eisenstein Series

Florin Spinu

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the

Department of Mathematics

November 2003

c© Copyright by Florin Spinu, 2005.

All Rights Reserved

Abstract

Let X = SL(2,Z)\H be the modular surface. We consider the Eisenstein series

with unitary parameter E(z, 12+it). We show that, when restricted to a fixed compact

subset Ω ⊂ X, the L4 norm ‖E(12

+ it)‖L4(Ω) is O(√

log t). On the other hand, it

is known from the work of Luo and Sarnak that ‖E(12

+ it)‖L2(Ω) is asymptotically

equal to cΩ

√log t. This shows that, in the continuous spectrum, the (generalized)

eigenfunctions of the hyperbolic Laplace operator have bounded L4 norm in the high

energy limit, after an appropriate normalization. This is in accord with the conjec-

tured behavior of eigenfunctions in the quantization of a classically chaotic system.

In the case of an arithmetic surface we reduce the L4 norm problem, via triple

product identities, to questions about a family sum of automorphic L-functions; tech-

niques from analytic number theory can then be applied successfully to establish a

sharp estimate for the L4 norm.

iii

Acknowledgements

I would like to thank my advisor, Peter Sarnak, for being a most inspiring teacher.

None of this work would have been possible without his guidance and encouragement.

I am grateful to him for being so generous in sharing his ideas (and time).

I am indebted to my friend Gergely Harcos for interesting mathematical conversa-

tions; his approximate functional equation spared me a lot of trouble. Many thanks

to Stephen D. Miller for helpful comments and for having the patience to read this

thesis.

My family deserves special acknowledgement for providing me with constant sup-

port. Special thanks to Laura for beating the computer at spotting mistakes.

iv

To My Parents, Doina and Marin

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 10

2.1 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Discrete spectrum: Hecke-Maass forms . . . . . . . . . . . . . 11

2.1.2 Continuous spectrum: Eisenstein series . . . . . . . . . . . . . 14

2.1.3 Parseval’s identity . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Truncated Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Maass-Selberg Relations and Normalization . . . . . . . . . . . . . . 17

2.3.1 The Random Wave Conjecture . . . . . . . . . . . . . . . . . 23

2.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Continuous Spectrum Contribution 27

3.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 An Explicit Formula for cA(T, t) . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 The terms R,M and T . . . . . . . . . . . . . . . . . . . . . 30

vi

3.3 The Continuous Spectrum Contribution . . . . . . . . . . . . . . . . 33

3.4 Main Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Riemann zeta-function and the gamma-function . . . . . . . . 34

3.4.2 Brief excursion into moments . . . . . . . . . . . . . . . . . . 36

3.5 An Estimate for∫ |R|2dt . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 An Estimate for∫ |M|2dt . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 An Estimate for∫ |T |2dt . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7.1 Further reduction of T1 . . . . . . . . . . . . . . . . . . . . . . 45

3.7.2 An estimate for the contribution of residues . . . . . . . . . . 47

3.8 An Estimate for∫∞

0|T ∗(T, t)|2dt . . . . . . . . . . . . . . . . . . . . 50

3.8.1 The integral∫ 4T

0|g1(t)|2dt . . . . . . . . . . . . . . . . . . . . 52

3.8.2 The integral∫∞4T|g1(t)|2dt . . . . . . . . . . . . . . . . . . . . 55


0|g2(t)|2dt . . . . . . . . . . . . . . . . . . . . 58

3.8.4 The integral∫∞4T|g2(t)|2dt . . . . . . . . . . . . . . . . . . . . 60

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Discrete Spectrum Contribution 68

4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 An Arithmetic Substitute . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 The formula of Luo and Sarnak . . . . . . . . . . . . . . . . . 70

4.2.2 Controlling the difference . . . . . . . . . . . . . . . . . . . . . 72

4.3 An Estimate for ‖H‖2 . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Computation of residues . . . . . . . . . . . . . . . . . . . . . 77

4.3.2 Evaluation of the shifted integral . . . . . . . . . . . . . . . . 79

4.4 An Estimate for ζ′′ζ

(1 + it) . . . . . . . . . . . . . . . . . . . . . . . . 81

5 A Family Sum 85

5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vii

5.1.1 An asymptotic formula for the weight . . . . . . . . . . . . . . 86

5.1.2 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.3 Decomposition after suitable ranges . . . . . . . . . . . . . . . 87

5.2 Main Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.1 Approximate functional equation . . . . . . . . . . . . . . . . 89

5.2.2 Kuznetsov’s trace formula . . . . . . . . . . . . . . . . . . . . 91

5.2.3 Spectral large sieve . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.4 Spectral second moment . . . . . . . . . . . . . . . . . . . . . 92

5.3 The Bulk Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.1 Part 1:∑

H≤tφ≤2H L4(12, φ) . . . . . . . . . . . . . . . . . . . . 94

5.3.2 Part 2:∑

H≤tφ≤2H

∣∣L(12

+ iT, φ)∣∣4 . . . . . . . . . . . . . . . . 97

5.3.3 Average over a finite interval . . . . . . . . . . . . . . . . . . . 101

5.4 The Transition Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Approximate formulae . . . . . . . . . . . . . . . . . . . . . . 104

5.4.2 Error term estimate . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.3 Leveling the argument . . . . . . . . . . . . . . . . . . . . . . 107

5.4.4 Lengthening the summation . . . . . . . . . . . . . . . . . . . 108

5.4.5 Applying the trace formula . . . . . . . . . . . . . . . . . . . . 111

5.4.6 Diagonal term . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.7 Nondiagonal term: sum of Kloosterman sums . . . . . . . . . 113

5.4.8 Nondiagonal term: continuous spectrum . . . . . . . . . . . . 114

5.4.9 Lemma on the Bessel transform . . . . . . . . . . . . . . . . . 117

5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6 Appendix 124

6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.1.1 The family sum . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2 Bulk Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

viii

6.2.1 Further reduction of the bulk sum . . . . . . . . . . . . . . . . 127

6.3 Diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3.1 The term I(hT ) . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3.2 The term J . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 Non-Diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.1 Integral Bessel Transform . . . . . . . . . . . . . . . . . . . . 133

6.4.2 Voronoi summation formula . . . . . . . . . . . . . . . . . . . 136

6.4.3 Analysis of S0(c, x) and S2(c, x) . . . . . . . . . . . . . . . . . 137

6.4.4 Analysis of S1(c, x) . . . . . . . . . . . . . . . . . . . . . . . . 137

6.5 Family Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.5.1 Diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.5.2 Off-diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5.3 The Additive Divisor Problem . . . . . . . . . . . . . . . . . . 148

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

ix

Chapter 1

Introduction

1.1 Thesis

Consider the full modular group Γ = SL(2,Z) and the associated modular surface

X = Γ\H, where H is the Poincare upper-half plane. X inherits the hyperbolic metric

of constant negative curvature (K ≡ −1) and is barely non-compact, having finite

volume and a cusp at infinity. In particular, the hyperbolic Laplacian ∆ has both

discrete and continuous spectrum in L2(X). The discrete spectrum is spanned by

the Hecke-Maass cusp forms, which are joint L2-eigenfunctions of ∆ and the Hecke

operators, while the continuous spectrum is furnished by the Eisenstein series with

unitary parameter E(z, 12

+ it), t ∈ R. These are generalized eigenfunctions (of both

∆ and the Hecke operators) of corresponding Laplace eigenvalue 14+t2. To render the

Eisenstein series in L2, we have to localize it to a compact regular domain Ω in X.

The asymptotic value of the L2 norm was computed by Luo and Sarnak [L-S, eq.2]

1

vol(Ω)

∫

Ω

∣∣E(z,1

2+ it)

∣∣2dz ∼ 2 log t

vol(X), t →∞ (1.1)

where dz = y−2dxdy is the hyperbolic volume element. We will show how this formula

can be derived from the Maass-Selberg relations, in the special case when Ω is a

1

truncated fundamental domain.

In the present thesis we study the behavior of the L4 norm of the Eisenstein series

in the high energy limit. Specifically, we prove

Theorem 1.1 (A). For a fixed Ω and an arbitrary ε > 0,

∫

Ω

∣∣E(z,1

2+ iT )

∣∣4dz = O(T ε), T →∞

The implied constant depends on Ω and ε.

The proof of this theorem occupies most of the thesis. In the Appendix we outline

the key steps for proving the following stronger result:

Theorem 1.2 (B).

∫

Ω

∣∣E(z,1

2+ iT )

∣∣4dz = O(log2 T ), T →∞

with the implied constant depending on Ω.

Remark 1.1. The meaning of Theorem B is that the restriction of the L2 normalized

Eisenstein series

E(z,1

2+ iT ) :=

E(z, 12

+ iT )√2 log T

(1.2)

to a (fixed) compact domain has bounded L4-norm as T →∞.

1.2 Motivation

The main motivation for our thesis is the Lp norm problem for the Laplace eigen-

functions on a negatively curved surface.

Let X be a (compact) surface endowed with a Riemannian metric of negative

curvature, K < 0. It is a known fact (due to E. Hopf [H]) that the geodesic flow Gt

2

of X is ergodic, in the sense that almost all geodesics become equidistributed in time

with respect to the volume element dvol induced by the metric. Moreover, almost

every long orbit of Gt in the unit tangent bundle S1X is equidistributed with respect

to the Liouville measure dν.

The analysis of the quantization of the geodesic flow Gt reduces to the eigenvalue

problem of the Laplacian

∆ψ = λψ

Let ψλ be an L2-normalized eigenfunction corresponding to an eigenvalue λ, ‖ψλ‖2 =

1. (We assume that the eigenvalues are in increasing order such that 0 ≤ λ ↑ ∞.)

Iwaniec and Sarnak [Iw-Sa2] formulated the following general conjecture:

Conjecture 1.3. Let X be a (compact) surface of negative curvature. Then, for any

2 ≤ p ≤ ∞ and ε > 0,

‖ψλ‖p = Oε,p(λε), λ →∞

This conjecture is wide open and no results beyond the local estimates (described

below) are known in the case of an arbitrary surface of variable curvature. In fact,

for p = ∞ and X the modular curve, the conjecture implies the Lindelof hypothesis

(in the t aspect) for the Dedekind zeta function of an imaginary quadratic number

field. Theorem A however establishes this conjecture in the special case when X is

arithmetic, p = 4, and ψλ is an Eisenstein series, while Theorem B states the uniform

boundedness of the L4 norm in this particular case.

Local estimates

A theorem of C. Sogge [So] gives local estimates for the Lp norms of an eigenfunction of

a general elliptic differential operator on a Riemannian manifold M . When dim M =

3

2, this estimate gives

‖ψλ‖p =

O(λ18− 1

4p ), 2 ≤ p ≤ 6

O(λ14− 1

p ), 6 ≤ p ≤ ∞(1.3)

with ψλ a normalized eigenfunction of the Laplace operator. In particular, for p = 4

this reads

‖ψλ‖4 = O(λ116 ) (1.4)

These estimates are sharp for the sphere with the canonical metric (S2, can). They

are not sharp in the quantum chaos regime (negative curvature), especially in view

of Conjecture 1.3. In analogy with the estimates for L-functions, we can regard 1.3

as a convexity estimate for the Lp norm.

An additional motivation for our problem is the connection with the Random

Wave Conjecture and the Quantum Unique Ergodicity Conjecture.

Random Wave Conjecture

Originally formulated by Berry [Be] for quantizations of chaotic Hamiltonians, this

conjecture predicts that, in the case of negative curvature, the Laplace eigenfunctions

ψλ tend to exhibit Gaussian random behavior in the high energy limit. This conjecture

was extended by Hejhal and Rackner [H-R] to non-compact surfaces of finite volume.

In particular, they gave convincing numerical evidence when X = SL(2,Z)\H and

ψλ is an Eisenstein series or a cusp form.

The moment version of this conjecture asserts that, for 2 ≤ p < ∞ an integer,

and Ω ⊂ X a compact domain,

1

vol(Ω)

∫

Ω

ψpλdvol → σpcp, λ →∞

4

where cp is the pth moment of the normal distribution N(0, 1) and σ2 = vol(X)−1 is

the conjectured variance of the random wave. For p = 2 and p = 4 this is

1

vol(Ω)

∫

Ω

ψ2λdvol → 1

vol(X)(1.5)

1

vol(Ω)

∫

Ω

ψ4λdvol → 3

vol(X)2(1.6)

When ψλ is the normalized Eisenstein series, the first condition is a consequence of

(1.1); in the same context, even though it does not establish convergence, Theorem

B shows at least that the left-hand side of (1.6) is a bounded sequence.

Quantum Unique Ergodicity

We return to the general case of a (compact) surface of negative curvature. To each

normalized eigenfunction of the Laplacian we associate the probability measure

dµλ = |ψλ|2dvol

We call dµλ a quantum measure; as is well known, this measure gives the probability

distribution of a particle in the state ψλ.

A theorem of Shnirelman, Zelditch and Colin de Verdiere states the existence of

a full density subsequence of eigenvalues λn, such that

dµn → dvol (1.7)

in the weak topology. This statement establishes ergodicity at the quantum level, in

the sense that almost all quantum measures become uniformly distributed in the high

energy limit λ → ∞. Zelditch extended this result to the case when X = Γ\H is a

non-compact surface of finite volume (Γ = Fuchsian).

The Quantum Unique Ergodicity Conjecture (QUE), as formulated by Rudnick

5

and Sarnak [R-S], states that there are no exceptional subsequences to property (1.7);

in other words, that all quantum measures dµλ become equidistributed in the high

energy limit: limλ→∞

dµλ = dvol; hence the term ’unique ergodicity’. We remark that

the right space to define this conjecture is S1X, where the quantum measures are

defined by microlocal lifts. The uniform measure is the Liouville measure.

Remark. Unlike in the classical situation, where periodic orbits of Gt (exceptions

to ergodicity) usually exist in abundance, at the quantum level, according to this

conjecture, such a phenomenon does not happen: weak limits of quantum measures

are not supposed to concentrate on a closed geodesic. In fact, it is clear that if ψλ

have uniformly bounded L4-norm, then any weak limit of the quantum measures is in

L2; therefore it cannot be supported in a lower dimensional submanifold. Moreover,

uniform boundedness of the L4 norm of the microlocal lifts to S1X implies the QUE

conjecture.

Accordingly, the property of QUE for the Eisenstein series

(QUE) for f ∈ Cc(X), T →∞,∫

X

f(z)∣∣E(z,

1

2+ iT )

∣∣2dz ∼ 2 log T

vol(X)

∫

X

f(z)dz (1.8)

can essentially be regarded as a consequence of Theorem B. This property was proved

by Luo and Sarnak [L-S].

QUE for the cusp forms of SL(2,Z)\H is still an open problem, though Linden-

strauss [Li] has made substantial progress using new methods from ergodic theory.

Theorem B gives evidence that an approach based on the uniform boundedness of the

L4 norm might work in this case as well. Sarnak and Watson [Sa-Wa] have recently

established Theorem A for Hecke-Maass cusp forms, under some mild hypotheses.

6

1.3 Proof

We return to the case where X is the modular surface. A variant of the triple product

formula, as given in [L-S], expresses the inner product⟨E2, φ

⟩between the square of

an Eisenstein series and a Hecke-Maass form in terms of automorphic L-functions.

By Parseval’s identity, the L4 norm of the Eisenstein series on a compact subset Ω

can be reduced to a weighted sum of automorphic L-functions (‘family sum’)

∥∥E(1

2+ iT )

∥∥4

L4(Ω)≈

∑

φ

w(tφ, 2T )L2(1

2, φ)

∣∣L(1

2+ 2iT, φ)

∣∣2 (1.9)

where φ ranges over the entire countable family of L2-normalized Hecke-Maass forms.

The various error terms arising from localization and the continuous spectrum con-

tribution are of admissible order for the intended purpose.

The analysis of this family sum splits naturally into two parts: the bulk, on which

an appropriate use of the Deshouillers-Iwaniec spectral large sieve inequality produces

the upper bound O(T ε), and a transitional range (see Chapter 5) where we employ

Kuznetsov’s trace formula.

In order to prove Theorem B, we then need to prove the estimate (“Lindelof on

average”)

1

T 2

∑

θT≤tφ≤2(1−θ)T

αφL2(

1

2, φ)

∣∣L(1

2+ 2iT, φ)

∣∣2 ¿ |ζ(1 + 2iT )|4 log2 T (1.10)

where 0 < θ < 1 is an arbitrarily small positive number, and αφ is a natural normaliz-

ing factor. (The transition range (1− θ)T ≤ tφ ≤ T requires separate discussion; the

details are presented in Chapter 5.) By means of the approximate functional equation

and Kuznetsov’s trace formula, we convert the left-hand side into a diagonal term

giving the main contribution, and a non-diagonal (sum of Kloosterman sums), which

is essentially of a lower order of magnitude. To this end, we open the Kloosterman

7

sums

S(m,n; c) =∑

x(mod c)∗e(mx + nx

c

)

and apply the Voronoi summation formula for the divisor function. The problem is

in this way reduced to that of seeking power saving cancellation in the shifted divisor

sum (smooth):

Sh(T ) :=∑

n∼T 2

τiT (n)τ(n + h) ¿ T 2−α (1.11)

uniformly in |h| ¿ T ε, for some α > 0.

Note that the spectral parameter T survives a first application of the trace formula.

To obtain cancellation in the shifted divisor sum, we have to further embed this

sum in a larger family and seek cancellation in a short sum, as in [Sa2]. By positivity,

the estimate (1.11) is certainly implied by the estimate

∑T≤tφ≤T+G

∣∣Sh(φ)∣∣2 +

∫ T+G

T

∣∣Sh(t)∣∣2dt ¿ T 3G (1.12)

for G = T 1−2δ, where the quantity

Sh(φ) =∑

n∼t2φ

λφ(n)τ(n + h)

was constructed by analogy with Sh(t): τit(n) are the Fourier coefficients of the unitary

Eisenstein series, while λφ(n) are the Fourier coefficients of the Maass-Hecke cusp form

φ. To prove (1.12), we open up the parentheses and apply the trace formula once

again. This reduces the problem to the analysis of another non-diagonal quantity

∑

c≤T δ

∑

m∼T 2

∑

n∼T 2

τ(m)τ(n)S(m,n; c)

8

with a smooth weight. The gain is that the spectral parameter T is finally cleared

from the argument of the divisor function. We open up the Kloosterman sums once

again and apply Voronoi’s formula for the sum in τ(m): this restricts the range of

m,n to near the diagonal.

Finally, we use available results from the classical additive divisor problem. Ini-

tiated by Ingham [In2] and Estermann [E], the asymptotic formula for the divisor

sum

Th(X) :=∑n≤X

τ(n)τ(n + h) (1.13)

has an extensive literature. Specifically, we use a formula of Motohashi [Mot2]. The

details are presented in the Appendix.

We end this section with the remark that, unlike in the traditional applications

(viz. subconvexity), the family method yields in our case an estimate which is sharp:

that is, the L4 norm estimate for the Eisenstein series. As with most applications of

the family method in GL2, the analysis of shifted convolutions of Fourier coefficients

is central to our approach.

9

Chapter 2

Preliminaries

2.1 Spectral Theory

Let Γ = SL(2,Z) and X = Γ\H. The hyperbolic metric ds2 = y−2(dx2+dy2

)induces

the volume element dz = y−2dxdy. With these, X has finite volume: vol(X) = π3.

The corresponding fundamental domain is

F := z ∈ H∣∣|z| ≥ 1, |x| ≤ 1/2

X is non-compact, having a cusp which corresponds to the point i∞ in the funda-

mental domain. The hyperbolic Laplacian

∆ = −y2( ∂2

∂x2+

∂2

∂y2

)

is a positive, self-adjoint, unbounded operator on L2(X). Under its action, L2(X) is

a direct sum of closed, infinite-dimensional, subspaces

L2(X) = L2d ⊕ L2

c

10

such that ∆ has discrete spectrum on L2d and purely continuous spectrum on L2

c .

The Hecke operators Tn are explicitly defined by

Tnf(z) =1√n

∑n=ada,d≥1

∑

b(mod d)

f(az + b

d

), n ≥ 1

To this, we add the ”symmetry”

T−1f(z) = f(−z)

and let T∞ = ∆. Then Tn−1≤n≤∞ is a commuting family of self-adjoint operators

which preserves the subspaces L2d and L2

c .

2.1.1 Discrete spectrum: Hecke-Maass forms

It follows that L2d has an orthonormal basis which consists of the constant function

φ0 = (vol(X))−1/2 and the L2-normalized, joint eigenfunctions of all Tn:

∆φ = λφφ

Tnφ = λφ(n)φ, n ≥ 1

T−1φ = εφφ, εφ = ±1

(2.1)

These are the Hecke-Maass forms. The ones with εφ = 1 are even, while those with

εφ = −1 are odd. In the present case, it is a consequence of λφ ≥ 14

(φ 6= φ0) that

these forms also satisfy the cuspidality condition:

∫ 1

0

φ(x + iy)dx = 0, y > 0

11

The fact that the space L2d is infinite-dimensional is highly non-trivial. Weyl’s law

N(λ) :=∑

0<λφ≤λ

1 =vol(X)

4πλ + O

(√λ log λ

)(2.2)

is a direct consequence of Selberg’s celebrated trace formula. In practice, it is conve-

nient to write λφ = 14

+ t2φ, with tφ > 0 whenever λφ > 14.

A theorem of Hecke identifies the Hecke eigenvalues with the coefficients of the

Fourier expansion; we have the identity (Fourier expansion)

φ(z) = ρφ(1)∑

n6=0

λφ(n)√

yKitφ(2π|n|y)e(nx) (2.3)

with λφ(−n) = εφλφ(n) and ρφ(1) a normalizing factor ensuring ‖φ‖2 = 1. The

K-Bessel function is given explicitly by

Kα(y) =1

2

∫ ∞

0

e−y2(t+t−1)tα

dt

t

Remark 2.1. The abundance of the discrete spectrum is a feature of the arithmetic-

ity of Γ, which translates into the existence of an infinite family of symmetries, or

correspondences (Hecke operators). Work of Phillips and Sarnak [Ph-S] gives evi-

dence that the discrete spectrum of a non-compact generic surface might in fact be

finite.

Hecke L-functions

The L-function associated to a Hecke-Maass form is the Dirichlet series

L(s, φ) :=∞∑

n=1

λφ(n)

ns

12

This series converges absolutely in a right half-plane where it clearly defines a holo-

morphic function. It is a well-known fact that L(s, φ) admits an analytic continuation

to the complex plane s ∈ C and defines an entire function satisfying the functional

equation

Λ(s, φ) := π−sΓ(s +

1−εφ

2+ itφ

2)Γ(

s +1−εφ

2− itφ

2)L(s, φ) = εφΛ(1− s, φ)

Euler product

As eigenvalues of the Hecke operators Tn, the coefficients λφ(n) satisfy the Hecke

relations

λφ(m)λφ(n) =∑

d|(m,n)

λφ

(mn

d2

)(2.4)

where (m,n) denotes the greatest common divisor. This translates into the Euler

product factorization of the L-function

L(s, φ) =∏

p

(1− λφ(p)p−s + p−2s

)−1, <s > 1 (2.5)

where the product is over all primes p.

Normalizing factor

The normalizing factor ρφ(1) is related to another automorphic L-function, the sym-

metric square L-function:

αφ :=|ρφ(1)|2

cosh(πtφ)=

2

L(1, sym2 φ)

Results of Iwaniec [Iw4] and Hoffstein-Lockhart [Hof-Lo] ensure that

t−εφ ¿ αφ ¿ tεφ (2.6)

13

for an arbitrarily small ε. This is an important fact that will be used throughout the

present thesis.

2.1.2 Continuous spectrum: Eisenstein series

The series

E(z, s) =∑

Γ∞\Γy(γz)s, <s À 0

is absolutely convergent in a right half-plane where it defines an automorphic function

in z ∈ X such that ∆E(z, s) = s(1− s)E(z, s).

A theorem of Selberg states that E(z, s) admits an analytic continuation to the

entire complex plane s ∈ C; moreover, E(z, s) is regular on the line <s = 12. This

fact holds in greater generality, when Γ is a Fuchsian group of the first kind. It is also

known that the Eisenstein series with unitary parameter E(z, 12

+ it) span the space

L2c (direct integral).

In the case Γ = SL(2,Z), the Eisenstein series has the following Fourier expansion

E(z, s) = ys + φ(s)y1−s +2

ξ(2s)

∑

n6=0

τs−1/2(|n|)√yKs−1/2(2π|n|y)e(nx) (2.7)

where τα(n) =∑

n=d1d2dα

1d−α2 = n−ασ2α(n) is the generalized divisor sum, and ξ(s) =

π−s/2Γ( s2)ζ(s) is the completed zeta-function. The scattering function φ(s) can be

expressed in terms of ξ(s):

φ(s) =ξ(2s− 1)

ξ(2s)(2.8)

The constant term of the Eisenstein series is e(y, s) = ys + φ(s)y1−s.

Remark 2.2. The regularity of E(z, s) on the line <s = 12, combined with the explicit

Fourier expansion of the Eisenstein series for SL(2,Z), implies the non-vanishing of

ζ(s) on the line <s = 1: ζ(1 + it) 6= 0. This gives a spectral proof of the Prime

Number Theorem.

14

E(z, s) is also an eigenfunction of the Hecke operators

TnE(z, s) = τs−1/2(n)E(z, s)

so that the analogy with the Maass-Hecke cusp forms is complete. In particular, it

also follows that the divisor functions satisfy the Hecke relations

τs(m)τs(n) =∑

d|(m,n)

τs

(mn

d2

)(2.9)

The L-function associated to E(z, 12

+ it) is

∞∑n=1

τit(n)

ns= ζ(s + it)ζ(s− it)

2.1.3 Parseval’s identity

The Plancherel formula for the spectral decomposition of ∆ is the following identity

f(z) =∑

φ

⟨f, φ

⟩φ(z) +

1

4π

∫ ∞

−∞

⟨f, E(

1

2+ it)

⟩E(z,

1

2+ it)dt (2.10)

where f ∈ C∞(X) is a smooth function of rapid decay in the cusp.

Parseval’s identity gives the spectral expansion of the L2 norm:

‖f‖22 =

∑

φ

∣∣⟨f, φ⟩∣∣2 +

1

4π

∫ ∞

−∞

∣∣∣⟨f, E(

1

2+ it)

⟩∣∣∣2

dt (2.11)

Here, as well as in the previous sum, φ ranges over an orthonormal basis of the discrete

spectrum (φ0 together with the Hecke-Maass forms).

15

2.2 Truncated Eisenstein Series

Let A ≥ 1 be a fixed parameter and consider the compact subset of X

XA := z ∈ X∣∣max

γ∈Γy(γz) ≤ A

The corresponding fundamental domain is FA = z ∈ F∣∣y(z) ≤ A. The set CA =

F − FA represents ’the cusp’.

Since any compact subset Ω ⊂ X is included in XA, when A is large enough, it

follows that in order to prove

∫

Ω

∣∣E(z,1

2+ iT )

∣∣4dz = O(log2 T ) (2.12)

it is enough to consider only the case Ω = XA. To this end, we define the truncated

Eisenstein series:

EA(z, s) =

E(z, s), z ∈ FA

E(z, s)− e(y, s), z ∈ CA

The way it is defined, EA(z, s) is smooth outside the horocycle y(z) = A. More

importantly, it is rapidly decreasing in the cusp, and it remains an eigenfunction of

∆ in the domain of smoothness.

Since EA agrees with E on XA, any upper bound estimate satisfied by

∫

X

∣∣EA(z,1

2+ iT )

∣∣4dz (2.13)

is automatically satisfied by

∫

XA

∣∣E(z,1

2+ iT )

∣∣4dz (2.14)

16

We find it more convenient to work with the integral (2.13). This constitutes the

main object of our study.

2.3 Maass-Selberg Relations and Normalization

In this section we compute the L2 norm of a localized Eisenstein series. The polarized

version of this formula (the inner product of two truncated Eisenstein series of arbi-

trary complex parameters) is known under the name of the Maass-Selberg relations,

and is fundamental in the understanding of the spectrum of ∆ on Γ\H. The main

tool is the following theorem from calculus:

Theorem 2.1. (Flux-Divergence Theorem) Assume M is a Riemannian manifold

with boundary, and f, g are smooth functions on M . Then

∫

M

(∆fg − f∆g

)dvol =

∫

∂M

(f

∂g

∂n− g

∂f

∂n

)dσ

where the volume element (dvol) is induced by the Riemannian metric, while dσ is the

induced area element on the boundary. ~n is the unit normal pointing in the outward

direction.

We apply the theorem to the case when M = FA, with boundary the horocycle

∂FA = y = A, 0 ≤ x ≤ 1, and the functions f = E(z, s1) and g = E(z, s2). We

have

∫

FA

∆zE(z, s1)E(z, s2)dz −∫

FA

E(z, s1)∆zE(z, s2)dz

=

∫ 1

0

E(z, s1)∂E

∂y(z, s2)− ∂E

∂y(z, s1)E(z, s2)dx (2.15)

On the other hand, in the case when M = CA (same boundary as FA but reversed

17

orientation), and for f = EA(z, s1), g = EA(z, s2), we have

∫

CA

∆EA(z, s1)EA(z, s2)dz −∫

CA

EA(z, s1)∆EA(z, s2)dz

=

∫ 1

0

−EA(z, s1)∂EA

∂y(z, s2) +

∂EA

∂y(z, s1)EA(z, s2)dx (2.16)

By adding the two identities term by term, and taking into account that

∆E(z, s) = s(1− s)E(z, s) and ∆EA(z, s) = s(1− s)EA(z, s), (2.17)

we obtain

(s1(1− s1)− s2(1− s2)

) ∫

FEA(z, s1)EA(z, s2)dz

=

∫ 1

0

[E(z, s1)

∂E

∂y(z, s2)− EA(z, s1)

∂EA

∂y(z, s2)− ∂E

∂y(z, s1)E(z, s2)

+∂EA

∂y(z, s1)EA(z, s2)

]dx (2.18)

with the last integral on the horocycle y = A, 0 ≤ x ≤ 1.

Since E(z, s) = e(y, s) + EA(z, s), we can further write the right-hand side as

∫ 1

0

[e(y, s1)

∂e

∂y(y, s2) + e(y, s1)

∂EA

∂y(z, s2) + EA(z, s1)

∂e

∂y(y, s2)

− ∂e

∂y(y, s1)e(y, s2)− ∂e

∂y(y, s1)EA(z, s2)− ∂EA

∂y(z, s1)e(y, s2)

]dx

=

∫ 1

0

[e(y, s1)

∂e

∂y(y, s2)− ∂e

∂y(y, s1)e(y, s2)

]dx

The constant term is given explicitly by e(y, s) = ys + φ(s)y1−s, and the integral is

18

evaluated at y = A. The last expression equals

(ys1 + φ(s1)y

1−s1)(

s2ys2−1 + φ(s2)(1− s2)y

−s2)

− (s1y

s1−1 + φ(s1)(1− s1)y−s1

)(ys2 + φ(s2)y

1−s2)∣∣∣∣

y=A

= (s2 − s1)As1+s2−1 + (1− s1 − s2)φ(s2)A

s1−s2

+ (s1 − s2)φ(s1)φ(s2)A1−s1−s2 − (1− s1 − s2)φ(s1)A

s2−s1

Therefore

∫

FEA(z, s1)EA(z, s2)dz =

As1+s2−1 − φ(s1)φ(s2)A1−s1−s2

s1 + s2 − 1

+φ(s2)A

s1−s2 − φ(s1)As2−s1

s1 − s2

(2.19)

In particular, s1 = 12

+ δ + it and s2 = 12− it yields

∫

FEA(z,

1

2+ δ + it)EA(z,

1

2− it)dz =

Aδ − φ(12

+ δ + it)φ(12− it)A−δ

δ+

φ(12− it)Aδ+2it − φ(1

2+ δ + it)A−δ−2it

δ + 2it

Passing to the limit δ ↓ 0, we obtain

∫

X

∣∣EA(z,1

2+ it)

∣∣2dz = −φ′

φ(1

2+ it)+2 log A+

A2itφ(12− it)− A−2itφ(1

2+ it)

2it(2.20)

Since φ(s) = ξ(2−2s)ξ(2s)

= π2s−1 Γ(1−s)Γ(s)

ζ(2−2s)ζ(2s)

, we find that

−φ′

φ(1

2+ it) = 2

ξ′

ξ(1 + 2it) + 2

ξ′

ξ(1− 2it) (2.21)

Stirling’s asymptotic formula and Vinogradov’s estimate [Ti, 6.19.2]:

Γ′

Γ(1

2+ it) = log |t|+ O(1) and

ζ ′

ζ(1 + 2it) = O

((log t)2/3+ε

)

19

yield

ξ′

ξ(1 + 2it) = −1

2log π +

1

2

Γ′

Γ(1

2+ it) +

ζ ′

ζ(1 + 2it)

=1

2log t + O((log t)2/3+ε) (2.22)

−φ′

φ(1

2+ it) = 2 log t + O

((log t)2/3+ε

)(2.23)

Therefore

∥∥EA(1

2+ it)

∥∥2

2= 2 log t + 2 log A + O((log t)2/3+ε), t →∞ (2.24)

and the implied constant in the error term is absolute.

Integral in the cusp

In this section we compute the integral of∣∣EA(z, 1

2+ it)

∣∣2 in the cusp CA.

I :=

∫

CA

∣∣EA(z,1

2+ it)

∣∣2dz

=4

|ξ(1 + 2it)|2∫ 1

0

∫ ∞

A

∣∣∣∣∣∑

n 6=0

τit(n)√

yKit(2πny)e(nx)

∣∣∣∣∣

2dxdy

y2

=8

|ξ(1 + 2it)|2∫ ∞

A

∞∑n=1

τ 2it(n)K2

it(2πny)dy

y

=8

|ξ(1 + 2it)|2∞∑

n=1

τ 2it(n)g(2πnA) (2.25)

where

g(x) :=

∫ ∞

x

K2it(y)

dy

y(2.26)

Let

G(s) :=

∫ ∞

0

g(x)xs dx

x(2.27)

20

be the Mellin transform. Integrating by parts and using the Mellin-Barnes formula,

we obtain

G(s) =1

s

∫ ∞

0

xsK2it(x)

dx

x= 2s−3 Γ2( s

2)Γ( s

2+ it)Γ( s

2− it)

sΓ(s)

Inverting the Mellin transform we have

g(x) =1

2πi

∫

(10)

G(s)x−sds

and the integral is on the line <s = 10. The right-hand side of 2.25 becomes

I =8

|ξ(1 + 2it)|2∞∑

n=1

τ 2it(n)

1

2πi

∫

(10)

G(s)(2πnA)−sds

=8

|ξ(1 + 2it)|21

2πi

∫

(10)

G(s)(2πA)−s ·[ ∞∑

n=1

τ 2it(n)

ns

]ds (2.28)

Using Ramanujan’s identity [Ram]

∞∑n=1

τ 2it(n)

ns=

ζ2(s)ζ(s + 2it)ζ(s− 2it)

ζ(2s)

we can further write the integral I as

I =1

|ξ(1 + 2it)|2 ·1

2πi

∫

(10)

A−s ξ2(s)ξ(s + 2it)ξ(s− 2it)

sξ(2s)ds

=1

|ξ(1 + 2it)| ·[(sum of the residues) +

1

2πi

∫

(1/2)

ds]

(2.29)

after shifting the line of integration from <s = 10 to <s = 1/2. The only poles we

encounter are at s = 1 and s = 1± 2it; s = 1 is a double pole with residue

Ress=1 =a

A

∣∣ξ(1 + 2it)∣∣2 ·

[− log(eA) +

ξ′

ξ(1 + 2it) +

ξ′

ξ(1− 2it)

]+

b

A

∣∣ξ(1 + 2it)∣∣2

21

where a, b are the constants that occur in the power expansion near s = 1 :

ξ2(s)

ξ(2s)=

a

(s− 1)2+

b

s− 1+ · · · (2.30)

The poles at s = 1± 2it are simple and their residues are easier to evaluate

Ress=1±2it

|ξ(1 + 2it)|2 =A−(1±2it)

1± 2it· ξ2(1± 2it)

|ξ(1± 2it)|2 ·ξ(1± 4it)

ξ(2± 4it)

= O((At)−1 ξ(1± 4it)

ξ(2± 4it)

)= O(A−1t−3/2 log t) (2.31)

The sum of residues from 2.29 therefore satisfies

(sum of residues)

|ξ(1 + 2it)|2 =a

A· [− log(eA) + 2<ξ′

ξ(1 + 2it)

]+

b

A+ O(A−1t−3/2 log t) (2.32)

For an asymptotic formula, it remains only to determine the constant a. This is

a = 1ξ(2)

= 6π. On the other hand, the shifted integral from 2.29 is of lower order:

O(A− 12 T− 1

6 ) (its computation is almost identical to the one in section 4.3.2). Using

2.22 once again, we have, as t →∞,

∫

CA

∣∣EA(z,1

2+ it)

∣∣2dz =6

Aπlog t + O

(A−1(log t)2/3+ε

)+ O

(A−1 log A

)(2.33)

Conclusion

Eq. 2.33 and 2.24 combined yield

∫

XA

∣∣E(z,1

2+ it)

∣∣2dz ∼ (π

3− 1

A

) 6

πlog t, t →∞ (2.34)

Since vol(XA) = π3− 1

A, we conclude that

limt→∞

1

vol(XA)

∫

XA

∣∣E(z,1

2+ it)

∣∣2dz =1

vol(X)(2.35)

22

where E was defined at 1.2. This shows that E is L2-normalized on the special sets

Ω = XA. For a general compact set Ω, see (1.1).

2.3.1 The Random Wave Conjecture

Let ξ(1+2it)|ξ(1+2it)| = eiθ(t), t ∈ R. The Hejhal-Rackner [H-R] formulation of the Random

Wave Conjecture in the case X = SL(2,Z)\H predicts that the real-valued function

Ψt(z) := eiθ(t)E(z, 12

+ it)√2 log t

= eiθ(t)E(z,1

2+ it)

tends to Gaussian N(0, vol(X)−1/2

)in distribution, when restricted to an arbitrary

compact (regular) subset Ω ⊂ X. The moment formulation of this conjecture yields,

for p = 2 and p = 4, and t →∞

1

vol(Ω)

∫

Ω

∣∣E(z,1

2+ it)

∣∣2dz ∼ 6

πlog t (2.36)

1

vol(Ω)

∫

Ω

∣∣E(z,1

2+ it)

∣∣4dz ∼ 12( 3

π

)2log2 t (2.37)

We saw that the first relation is satisfied. The more subtle question concerning the

behavior of the L4 norm in the high energy regime will be addressed in the following

chapters. We will show how this problem can be reduced, in the arithmetic case

Γ = SL(2,Z), to a problem in the analytic theory of automorphic L-functions.

2.4 Structure of the Thesis

As was explained before, in order to prove Theorem A or B, it is enough to prove the

same estimate for the truncated Eisenstein series. Specifically, the main theorem of

the present thesis is

23

Theorem 2.2 (L4). For a fixed truncation parameter A and T →∞, we have

a) ‖EA(1

2+ iT )‖4 = O(T ε), ∀ε > 0

Moreover, the stronger result

b) ‖EA(1

2+ iT )‖4 = O

(√log T

)

holds true.

The proof of a) occupies most of the present thesis. Part b) constitutes the subject

of the Appendix.

The starting point is Parseval’s identity applied to the function E2A(z, 1

2+ iT ). We

have

‖EA(z,1

2+ iT )‖4

4 = Disc.(A; T ) + Cont.(A; T ) (2.38)

where Disc.(A : T ) is the discrete spectrum contribution

Disc.(A; T ) =∑

φ

∣∣∣⟨E2

A(1

2+ iT ), φ

⟩∣∣∣2

(2.39)

and Cont.(A; T ) represents the continuous spectrum contribution

Cont.(A; T ) =1

4π

∫

R

∣∣∣⟨E2

A(1

2+ iT ), E(

1

2+ it)

⟩∣∣∣2

dt (2.40)

The term with φ = φ0 in Disc.(A; T ) is bounded in absolute value by log2 T (this is

from the L2-norm computation). Hence

Disc.(A; T ) =∑

φ

∣∣∣⟨E2

A(1

2+ iT ), φ

⟩∣∣∣2

+ O(log2 T ) (2.41)

where now the sum is over the Hecke-Maass cusp forms. For the purpose of proving

24

Theorem L4, we can ignore from now on the error term in the formula of Disc.(A; T ).

The rest of the thesis is structured as follows:

i) In Chapter 3 we prove the existence of a positive number b > 0 such that

Cont.(A; T ) ≤ 108A + O(T−b)

The proof of this fact relies heavily on the fourth moment as well as a subconvexity

estimate for the Riemann zeta on the critical line <s = 12.

ii) In Chapter 4 we replace the discrete spectrum contribution by an arithmetic

quantity Disc.(∞; T ) (a weighted sum of automorphic L-functions), and show that

the difference is admissible:

∣∣∣Disc.(A; T )1/2 −Disc.(∞; T )1/2∣∣∣ = O(log T )

to the effect that

‖EA(z,1

2+ iT )‖4

4 ≤ 2 Disc.(∞; T ) + O(log2 T ) (2.42)

In this way, we establish a complete equivalence between the problem of estimating

the L4 norm and that of estimating the arithmetic quantity Disc.(∞; T ).

iii) Chapter 5 is devoted entirely to the analysis of Disc.(∞; T ). There, we prove

that

Disc.(∞; T ) = O(T ε), ∀ε > 0 (2.43)

This will accomplish part a) of Theorem L4.

iv) The Appendix is concerned with a finer analysis of the arithmetic quantity

Disc.(∞; T ). We outline the proof of the estimate

Disc.(∞; T ) = O(log2 T ) (2.44)

25

This will complete part b) of Theorem L4.

26

Chapter 3

Continuous Spectrum Contribution

3.1 Preliminary Remarks

The continuous spectrum contribution to the spectral expansion of the L4 norm of

the Eisenstein series EA(z, 12

+ iT ), is given explicitly by

Cont.(A; T ) =1

4π

∫ ∞

−∞|cA(T, t)|2dt (3.1)

with cA(T, t) representing the Fourier transform of E2A(1

2+ iT ) along the continuous

spectrum:

cA(T, t) :=

∫

X

E2A(z,

1

2+ iT )E(z,

1

2+ it)dz (3.2)

The aim of this chapter is to give an estimate, in the limit T →∞, for the quantity

Cont.(A; T ). The main result of the chapter is stated in Theorem 3.3.

3.2 An Explicit Formula for cA(T, t)

Lemma 3.1. Let F be an automorphic function with respect to Γ = SL(2,Z). Sup-

pose F has polynomial growth in y(z) in any vertical strip. When s lies in a right

27

half-plane, the following identity holds

∫

FA

F (z)E(z, s)dz +

∫

CA

F (z)(E(z, s)− ys

)dz =

∫ A

0

a0(y)ys−1dy

y

where a0(y) :=∫ 1

0F (x + iy)dx is the constant term of F .

Proof. Since s is in the region of absolute convergence of the Eisenstein series, we can

unfold the integral and obtain

∫

FF (z)(E(z, s)− ys)dz =

∑

γ∈Γ∞\(Γ−Γ∞)

∫

FF (z)ys(γz)dz

=∑

γ∈Γ∞\(Γ−Γ∞)

∫

γ(F)

ysF (z)dz =

∫

∪∗γ(F)

ysF (z)dz

and the star means that the union is over the coset representatives of the quotient

set Γ∞\(Γ− Γ∞). The domain of the last integral is thus ∪∗γγ(F) = F∞ − F . Since

A > 1, this coincides with [0, 1]× [0, A]−FA. Therefore

∫

FF (z)

(E(z,s)− ys

)dz =

∫ 1

0

∫ A

0

ysF (z)dxdy

y2−

∫

FA

ysF (z)dz

=

∫ A

0

a0(y)ys−1dy

y−

∫

FA

ysF (z)dz

Since F = FA ∪ CA, we can rewrite this identity as

∫

FA

F (z)(E(z,s)− ys)dz +

∫

CA

F (z)(E(z, s)− ys)dz =

∫

FF (z)

(E(z, s)− ys

)dz

=

∫ A

0

a0(y)ys−1dy

y−

∫

FA

ysF (z)dz

and this implies the statement of the lemma.

We apply the lemma to the automorphic function F (z) = E2(z, τ) where for the

time being, τ = 12+iT . By the general properties of the Eisenstein series, this function

28

has polynomial growth in any vertical strip. For <(s) > 1, we have

∫

FA

F (z)E(z, s)dz +

∫

CA

F (z)(E(z, s)− ys

)dz =

∫ A

0

a0(y)ys−1dy

y(3.3)

Here a0(y) is the constant term of E2(z, τ).

The second integral of 3.3 can be rewritten as:

∫

CA

F (z)(E(z, s)− ys

)dz =

∫

CA

F (z) · [E(z, s)− e(y, s) + φ(s)y1−s]dz

=

∫

CA

F (z)(E(z, s)− e(y, s)

)dz + φ(s)

∫

CA

y1−sF (z)dz

=

∫

CA

(F (z)− a0(y)

)E(z, s)dz + φ(s)

∫ ∞

A

a0(y)y−s dy

y

The last identity holds whenever <s is larger than the degree of the polynomial growth

of F (in the present case, <s > 1). For z ∈ CA, we have

F (z) = E(z, τ)2 =(EA(z, τ) + e(y, τ)

)2

and we can further rewrite the right-hand side as

∫

CA

[E2

A(z, τ) + 2e(y, τ)EA(z, τ) + e2(y, τ)− a0(y)]E(z, s)dz + φ(s)

∫ ∞

A

a0(y)y−s dy

y

=

∫

CA

[E2

A(z, τ) + 2e(y, τ)EA(z, τ)] · E(z, s)dz

−∫

CA

(a0(y)− e2(y, τ)

)e(y, s)dz + φ(s)

∫ ∞

A

a0(y)y−s dy

y

Equation 3.3 becomes

∫

FA

E2(s, τ)E(z, s)dz +

∫

CA

E2A(z, τ)E(z, s)dz +

∫

CA

2e(y, τ)EA(z, τ)E(z, s)dz

=

∫ A

0

a0(y)ys−1dy

y− φ(s)

∫ ∞

A

a0(y)y−s dy

y+

∫ ∞

A

(a0(y)− e2(y, τ))e(y, s)dy

y2

The first two terms of the left-hand side add up to∫F E2

A(z, τ)E(z, s)dz, while the

29

right-hand side can be rearranged so as to obtain

∫

FEA(z, τ)2E(z, s)dz =

∫ ∞

0

(a0(y)− e2(y, τ))ys−1dy

y

+

[∫ A

0

e2(y, τ)ys−1dy

y− φ(s)

∫ ∞

A

e2(y, τ)y−s dy

y

]

−∫

CA

2e(y, τ)EA(z, τ)E(z, s)dz

=: R+M−T (3.4)

3.2.1 The terms R,M and T

We now evaluate each of the terms arising in the identity 3.4.

The term R

Side remark: since a0(y) is the constant term of the function F = E(z, τ)2, the

quantity

R :=

∫ ∞

0

(a0(y)− e2(y, τ)

)ys−1dy

y(3.5)

can be interpreted, in terms of Zagier [Za], as the renormalization of a (divergent)

integral

R = R. N.

∫

X

E2(z, τ)E(z, s)dz

To actually compute this quantity we need an explicit formula for the constant term

a0(y) of E2(τ) :

a0(y) =

∫ 1

0

E2(x + iy, τ)dx

Using the explicit Fourier expansion 2.7 of the Eisenstein series, we obtain

a0(y) = e2(y, τ) +8y

ξ2(1 + 2iT )

∞∑n=1

τ 2iT (n)K2

iT (2πny)

30

The following identity then holds in the region of absolute convergence <s > 1 of the

Eisenstein series:

R =

∫ ∞

0

(a0(y)− e2(y, τ)

)ys−1dy

y

=

∫ ∞

0

8y

ξ2(1 + 2iT )

∞∑n=1

τ 2iT (n)K2

iT (2πny)ys−1dy

y

=8

ξ2(1 + 2iT )

∞∑n=1

τ 2iT (n)

(2πn)s

∫ ∞

0

ysK2iT (y)

dy

y(3.6)

Here we can use the Mellin-Barnes formula [G-R, 6.576]

∫ ∞

0

ysKµ(y)Kν(y)dy

y= 2s−3

∏±,± Γ

(s±µ±ν

2

)

Γ(s)(3.7)

and an identity of Ramanujan [Ti, 1.3]

∞∑n=1

σa(n)σb(n)

ns=

ζ(s)ζ(s− a)ζ(s− b)ζ(s− a− b)

ζ(2s− a− b)(3.8)

Then R admits an expression in terms of the Riemann zeta-function:

R =ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)ξ2(1 + 2iT )(3.9)

31

The term M

M =

∫ A

0

e(y, τ)2ys−1dy

y− φ(s)

∫ ∞

A

e(y, τ)2y−s dy

y

=

∫ A

0

(y1+2iT + 2yφ(τ) + φ2(τ)y1−2iT

)ys−1dy

y

− φ(s)

∫ ∞

A

(y1+2iT + 2φ(τ)y + φ2(τ)y1−2iT

)y−s dy

y

=As+2iT

s + 2iT+ 2φ(τ)

As

s+ φ2(τ)

As−2iT

s− 2iT

+ φ(s) ·[

A1−s+2iT

1− s + 2iT+ 2φ(τ)

A1−s

1− s+ φ2(τ)

A1−s−2iT

1− s− 2iT

](3.10)

Nothing more will be added about T at this point. This term will be treated in

greater detail later on. The next proposition collects the results obtained so far:

Proposition 3.2. For s ∈ C,

∫

X

EA(z,1

2+ iT )2E(z, s)dz = R+M−T (3.11)

where:

R =ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)ξ2(1 + 2iT )

M =As+2iT

s + 2iT+ 2φ(

1

2+ iT )

As

s+ φ2(

1

2+ iT )

As−2iT

s− 2iT+

+ φ(s) ·[

A1−s+2iT

1− s + 2iT+ 2φ(

1

2+ iT )

A1−s

1− s+ φ2(

1

2+ iT )

A1−s−2iT

1− s− 2iT

]

T =

∫

CA

2e(y,1

2+ iT )EA(z,

1

2+ iT )E(z, s)dz

Remark 3.1. The explicit formula for the inner product∫

XEA(z, τ)2E(z, s)dz, as

stated in Proposition 3.11, is valid initially in the range <(s) > 1 of absolute con-

vergence of the Eisenstein series. However, since the function EA(z, τ) is rapidly

32

decreasing in the cusp, while E(z, s) has polynomial growth, it follows that each

side of 3.11 defines a meromorphic function in s ∈ C. Regarded as an identity of

meromorphic functions, (3.11) must hold for all s ∈ C. In particular, we obtain an

identity when <(s) = 12, which is the case that gives cA(T, t). We remarked earlier

that the Eisenstein series is regular on the unitary line <s = 12. In particular, this

ensures that the Fourier transform cA(T, t) is a smooth, rapidly decreasing function

of t, which admits an explicit expression given by equation 3.11, if we let s = 12

+ it.

3.3 The Continuous Spectrum Contribution

Throughout this section we use the notation s = 12

+ it, t ∈ R. The quantities R,Mand T will then depend on the spectral parameter T and the variable t. They will be

referred to as M(T, t) etc. To simplify notation, we will not specify the dependence

on the truncation parameter A.

The previous proposition gives an expression for the Fourier transform cA(T, t),

initially defined at 3.2:

cA(T, t) = M(T, t) +R(T, t)−M(T, t) (3.12)

Using the Cauchy-Schwartz inequality we obtain a preliminary bound on the integral

of |cA(T, t)|2

Cont.(A; T ) =1

4π

∫ ∞

−∞|cA(T, t)|2dt

≤ 3

4π·∫ ∞

−∞|R(T, t)|2dt +

∫ ∞

−∞|M(T, t)|2dt +

∫ ∞

−∞|T (T, t)|2dt

(3.13)

The rest of the chapter is concerned with the proof of the following

33

Theorem 3.3. There exists a positive number b > 0 such that

Cont.(A; T ) ≤ 108A + O(T−b), T →∞

In particular, if the truncation parameter A is fixed, then Cont.(A; T ) = O(1), with

the implied constant depending on A only.

Proof. We shall prove that the integrals∫R |R(T, t)|2dt and

∫R |T (T, t)|2dt tend to

zero as T →∞, while∫R |M(T, t)|2dt is uniformly bounded as a function of T .

3.4 Main Ingredients

In this section we list a series of well-known properties of the Riemann zeta-function,

which play a crucial role in estimating the continuous spectrum contribution. The

main reference is E.C. Titchmarsh, The Theory of the Riemann Zeta-Function (re-

vised by D.R. Heath-Brown) [Ti].

3.4.1 Riemann zeta-function and the gamma-function

Stirling’s asymptotic formula

This formula gives the asymptotic behavior of Γ(s), when s = σ + it belongs to a

fixed vertical strip a ≤ σ ≤ b. For now, we are only interested in the absolute value:

|Γ(σ + it)| = e−12π|t||t|σ− 1

2

√2π1 + O(t−1), t →∞ (3.14)

We also need the asymptotic formula for the logarithmic derivative

Γ′

Γ(σ + it) = log t + O(1) (3.15)

valid in a vertical strip a ≤ σ ≤ b. Reference: [Ti, 4.12.2].

34

ζ(s) on the line <s = 1

(1 + |t|)−ε ¿ε ζ(1 + it) ¿ε (1 + |t|)ε, ∀ε > 0 (3.16)

The fractional power of the logarithm estimate: 1ζ(1+it)

= O((log t)23+ε) is due to

Vinogradov [Ti, 6.19]. However, we will not need this stronger form.

ζ(s) on the line <s = 12

The convexity estimate is obtained from the Phragmen-Lindelof principle which allows

us to interpolate trivial bounds for ζ(s) on the vertical lines <s = 1 and <s = 0. It

gives

ζ(1

2+ it) ¿ε (1 + |t|) 1

4+ε, ∀ε > 0 (3.17)

Any improvement on the exponent 14

gives a subconvex estimate. Such an estimate

should therefore be of the type

ζ(1

2+ it) ¿ε (1 + |t|)θ+ε, ∀ε > 0 (3.18)

where necessarily θ < 14. The first such improvement was achieved by Hardy and Lit-

tlewood in 1922, using Weyl’s method for estimating exponential sums. The exponent

obtained by this method is θ = 16, and the estimate

ζ(1

2+ it) ¿ε (1 + |t|) 1

6+ε (3.19)

is usually referred to as Weyl’s estimate. The exponent 16

was improved later on,

among others, by Titchmarsh [Ti, 5.3] who obtained θ = 27164

, using the method of

van der Corput. For the most recent bounds see [Hu].

35

Remark 3.2. We will only use the fact that

θ <1

6(3.20)

As an immediate consequence, we regard

ζ(12

+ it)

(1 + |t|)1/4= O(1) (3.21)

as the convexity bound.

We remark that the ideal subconvexity estimate is naturally given by the Lindelof

Hypothesis, which is the conjecture that θ = 0. The Lindelof Hypothesis is implied

by the Riemann Hypothesis, and in many applications these two have equal strength.

3.4.2 Brief excursion into moments

Efforts to understand the size of the Riemann zeta-function have led to the theory of

power moments.

Second moment

In 1926, Ingham [In] found an asymptotic formula for the mean square of the Riemann

zeta-function on the critical line:

∫ T

0

|ζ(1

2+ it)|2dt = T log T + (2γ − 1− log 2π)T + O(T

12 log T ) (3.22)

Later on, Balasubramanian [Ba] improved on this result by showing that the error

term is O(T13+ε), which is in accordance with Weyl’s 1

6subconvex estimate.

36

Fourth moment

Ingham also tackled the fourth moment, but he was able to give only the leading term

of the asymptotic formula:

∫ T

0

|ζ(1

2+ it)|4dt =

1

2π2T log4 T + O(T log3 T ) (3.23)

In 1979, Heath Brown [HB] found an asymptotic formula for the fourth moment of

the form TP4(log T )+Oε(T78+ε), where P4(x) is a polynomial of degree 4 with leading

coefficient (2π2)−1. In 1986, Zavorotnyi [Zav] proved that the size of the error term

is O(T23+ε), which again is in accordance with Weyl’s 1

6subconvex estimate.

Remark 3.3. We will use only the following average estimate on a long interval:

∫ T

0

|ζ(12

+ it)|41 + |t| dt = O(log5 T ), T →∞ (3.24)

This is a consequence of the following weaker form of 3.23:

∫ T

0

|ζ(1

2+ it)|4dt = O(T log4 T )

which is originally due to Hardy and Littlewood [H-L, 1922]. Moreover, even the

average estimate ∫ T

0

|ζ(12

+ it)|41 + |t| dt = O(T ε), ∀ε > 0 (3.25)

suffices for our purposes. As a matter of fact, this corresponds to Lemma ι in [H-L].

We sketch a proof of 3.25 which is based on an idea that is used again in Section

5.4. At the end of this proof we comment on the features that are common to both

situations.

The starting point is a formula of Hardy and Littlewood [H-L2] which approxi-

37

mates ζ2(s) on the critical line by a Dirichlet polynomial of finite length:

ζ2(1

2+ it) =

∑

n≤ tx2π

τ(n)

n1/2+it+ γt

∑

n≤ t2πx

τ(n)

n1/2−it+ O(log t) (3.26)

uniformly in x ³ 1, i.e. x bounded from below and above by an absolute constant.

Here τ(n) =∑

d|n 1 is the divisor function and |γt| = 1. We do not discuss the details

of this formula, since we reserve a special section for the concept of the approximate

functional equation (Section 5.2.1).

We note that the freedom provided by the extra parameter x allows us to reduce

the task of proving 3.25 to that of proving

I(t) :=

∫ T

T/2

∣∣∣∣∣∑n¿T

τ(n)

n1/2+it

∣∣∣∣∣

2

dt = O(T 1+ε) (3.27)

for any positive ε. Let ψ ≥ 0 be a positive, smooth function of compact support in

(−2, 2), such that ψ ≥ 1 on [−1, 1]. Then I(T ) is certainly less than

J :=

∫ ∞

−∞

∣∣∣∣∣∑n¿T

τ(n)

n1/2+it

∣∣∣∣∣

2

ψ( t

K

)dt (3.28)

where K = T 1+ε, ε > 0. Thus we extended the range of integration from t ≤ T in 3.27

to t ¿ K log2 T in 3.28. The advantage of having a smooth kernel in the definition

of J is that we can use the Fourier transform to obtain

J = K∑

m,n¿T

τ(m)τ(n)√mn

ψ( K

2πlog(m/n)

)(3.29)

We distinguish between two kinds of terms in this sum: a diagonal contribution

corresponding to m = n, and a non-diagonal corresponding to m 6= n: J = D + ND,

38

with

D = Kψ(0)∑m¿T

τ 2(m)

m¿ KT ε (3.30)

and

ND = K∑

m,n¿T,m6=n

τ(m)τ(n)√mn

ψ( K

2πlog(m/n)

)(3.31)

Suppose m > n. Then log(m/n) ≥ m−nn≥ 1

nÀ 1

T, and hence the argument of ψ is

K

2πlog(m/n) À K

T= T ε

Since ψ is a Schwartz function, ψ(

K2π

log(m/n))

= O(T−N) for any N ≥ 1, hence the

entire quantity ND is negligible. Therefore the main contribution to J is given by

D. By 3.30 we conclude that J , and therefore I(T ), is O(T 1+ε), for any positive ε.

This completes the proof of 3.25.

Remark 3.4. In many cases, an average (or moment) of a family of L-functions

can be reduced, after applying the trace formula, to a diagonal and a non-diagonal

component. In practice, the diagonal is the easier term to evaluate and yields the

‘main term’, while the non-diagonal, having extra parameters, is harder to manage.

We illustrated a way of eliminating the non-diagonal completely, by simply extending

the averaging family (over-summation) and exploiting positivity. The method works

at the expense of a slightly larger, yet still admissible, diagonal term. We shall use

the same idea once more in Section 5.4, where we obtain an average estimate for

values of Hecke L-functions.

3.5 An Estimate for∫ |R|2dt

In this section we prove the following estimate for the integral of R:

39

Proposition 3.4. There exists an absolute constant B1 > 0 such that the inequality

∫ ∞

−∞|R(T, t)|2dt ≤ B1T

−1/6 (3.32)

holds for T ≥ 1.

Proof. We recall the expression for R, eq. 3.9:

R(T, t) =ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)ξ2(1 + 2iT )

= π12+i(2T−t) · Γ2(1

4+ i

2t)Γ(1

4+ i

2(t + 2T ))Γ(1

4+ i

2(t− 2T ))

Γ(12

+ it)Γ2(12

+ iT )

×ζ2(12

+ it)ζ(12

+ i(t + 2T ))ζ(12

+ i(t− 2T ))

ζ(1 + 2it)ζ2(1 + 2iT )(3.33)

By Stirling’s formula (3.14), we bound the ratio of Gamma functions by

Gamma factor ¿ (1 + |t|)− 12 (1 + |t− 2T |)− 1

4 (1 + |t + 2T |)− 14

× expπT − π

4|t− 2T | − π

4|t + 2T | (3.34)

and we note that the exponential factor is always ≤ 1.

The next step is to split the R-integral into convenient ranges:

∫ ∞

0

|R|2dt =

∫ 3T

0

|R|2dt +

∫ ∞

3T

|R|2dt

Range 0 ≤ t < 3T

By 3.34, when t is in this range,

|R(T, t)| ¿ |ζ(12

+ it)|2(1 + |t|)1/2

|ζ(12

+ i(t + 2T ))|(1 + |t + 2T |)1/4

|ζ(12

+ i(t− 2T ))|(1 + |t− 2T |)1/4

1

|ζ(1 + 2it)ζ(1 + 2iT )|

40

We apply the subconvexity estimate 3.18 to the second ratio to find that it is bounded

by T− 14+θ+ε; from 3.21 we know that the third ratio is bounded by an absolute con-

stant; by 3.16, the last ratio is bounded by T ε. We obtain

∫ 3T

0

|R(T, t)|2dt ¿ T− 12+2θ+ε

∫ 3T

0

|ζ(12

+ it)|41 + |t| dt

We can now employ the fourth moment (3.24) of the Riemann zeta to see that the

last integral is bounded by log5 T . Hence the right-hand side is O(T− 12+2θ+ε). We

know from eq. 3.20 that θ < 16. Therefore

∫ 3T

0

|R(T, t)|2dt = O(T−1/6) (3.35)

Range 3T ≤ t < ∞

It can be checked very easily that the integral∫∞3T|R(T, t)|2dt has an exponential

decay as a function of T , when T → ∞. For t > 3T , the Gamma factor in Ris bounded, in view of 3.34, by t−1 exp

(−π2t + πT

); the factors of zeta multiplied

together are no larger in absolute value than |t|2. The integral∫∞3T|R(T, t)|2dt is thus

bounded, up to an absolute constant, by

∫ ∞

3T

t2 exp (−πt + 2πT ) ¿ exp(−πT ) (3.36)

This finishes the proof of Proposition 3.4.

3.6 An Estimate for∫ |M|2dt

It is a well known fact in the general theory of Eisenstein series that the scattering

function φ(s) is regular and unitary on the line <s = 12. In our case, φ(s) has an

explicit formula given by 2.8. The fact that |φ(12+ it)| = 1 follows from the functional

41

equation of the Riemann zeta ξ(s) = ξ(1−s), together with the fact that ξ(s) = ξ(s).

Therefore when t ∈ R,

φ(1

2+ it) =

ξ(1− 2it)

ξ(1 + 2it)⇒ |φ(

1

2+ it)| ≡ 1

We can then obtain an upper bound for M(T, t) directly from 3.10 :

|M(T, t)| ≤ 2√

A ·[

1

|12

+ i(t + 2T )| +2

|12

+ it| +1

|12

+ i(t− 2T )|]

The Cauchy-Schwartz inequality and the relation∫∞−∞

dt1/4+t2

= 2π imply

Proposition 3.5. For T ≥ 0,

∫ ∞

−∞|M(T, t)|2dt ≤ 144πA (3.37)

3.7 An Estimate for∫ |T |2dt

The expression of T is slightly more complicated than that of R or M. However, the

ingredients used to estimate the integral of T are the same as those already used in

the case of R: a subconvex estimate and the fourth moment for ζ(s) on the critical

line. The goal of this section is to prove the following

Proposition 3.6. There exists an absolute constant B2 such that

∫ ∞

−∞|T (T, t)|2dt ≤ B2T

−1/6 (3.38)

holds for any T ≥ 1.

We recall the expression for T (T, t) from Proposition 3.2:

T (T, t) =

∫

CA

2e(y,1

2+ iT )EA(z,

1

2+ iT )E(z,

1

2+ it)dz (3.39)

42

Using the explicit Fourier expansion of the Eisenstein series from 2.7, we can further

write this term as

T (T, t) =16

ξ(1 + 2iT )ξ(1 + 2it)

∞∑n=1

τiT (n)τit(n)×

×∫ ∞

A

e(y,1

2+ iT )KiT (2πny)Kit(2πny)

dy

y

The constant term of the Eisenstein series is e(y, 12

+ iT ) = y12+iT + φ(1

2+ iT )y

12−iT .

Therefore, the integral 3.39 splits as a sum of two integrals, and correspondingly Twill be a sum of two terms: T (T, t) = T1(T, t) + T2(T, t). Once again, the unitarity

of φ(s) on the critical line <s = 12

ensures that |T1(T, t)| = |T2(T, t)|. Combined with

the relation |T1(T, t)| = |T1(T,−t)| and the Cauchy Schwartz inequality, this leads to

∫ ∞

−∞|T (T, t)|2dt ≤ 8

∫ ∞

0

|T1(T, t)|2 (3.40)

where

T1(T, t) =16

ξ(1 + 2iT )ξ(1 + 2it)

∞∑n=1

τiT (n)τit(n)×

×∫ ∞

A

y12+iT KiT (2πny)Kit(2πny)

dy

y(3.41)

In view of the previous inequality, the statement of Proposition 3.6 is equivalent to

∫ ∞

0

|T1(T, t)|2dt = O(T−1/6) (3.42)

A change of variable in 3.41 yields

T1(T, t) =16

ξ(2τ)ξ(2s)

∞∑n=1

τiT (n)τit(n)

(2πn)12+iT

∫ ∞

2πnA

y12+iT KiT (y)Kit(y)

dy

y

=16

ξ(2τ)ξ(2s)

∞∑n=1

τiT (n)τit(n)

(2πn)12+iT

g(2πnA) (3.43)

43

where g(x) stands for the integral expression involving the K-Bessel functions:

g(x) =

∫ ∞

x

y12+iT KiT (y)Kit(y)

dy

y(3.44)

The function g(x)

Equation 3.44 gives an immediate expression for the derivative g′(x). We can then

compute the Mellin transform of g(x) itself using integration by parts first, and then

the Mellin-Barnes formula (3.7). Let G(s) be the Mellin transform of g(x).

G(s) :=

∫ ∞

0

g(x)xs dx

x= −1

s

∫ ∞

0

xs+1g′(x)dx

x

=1

s

∫ ∞

0

xs+ 12+iT KiT (x)Kit(x)

dx

x

=2s+ 1

2+iT−3

s·∏± Γ

( s+ 12±it

2

) ∏± Γ

( s+ 12±it

2+ iT

)

Γ(s + 12

+ iT )(3.45)

By the Mellin inversion formula, we obtain a convenient integral expression for g(x):

g(x) =1

2πi

∫

(σ0)

x−sG(s)ds

where the integral is over the vertical line <s = σ0. This identity is valid for σ0 > 1.

44

3.7.1 Further reduction of T1

We return now to eq. 3.43 which gives an expression for T1. To simplify the writing,

we ignore for the moment the factor 16ξ(2τ)ξ(2s)

. Then:

T1(T, t) =∞∑

n=1

τiT (n)τit(n)

(2πn)12+iT

g(2πnA)

=∞∑

n=1

τiT (n)τit(n)

(2πn)12+iT

· 1

2πi

∫

(σ0)

(2πnA)−sG(s)ds

=A

12+iT

2πi

∫

(σ0)

(2πA)−(s+ 12+iT )G(s) ·

[ ∞∑n=1

τiT (n)τit(n)

ns+ 12+iT

]ds

=A

12+iT

2πi

∫

(σ0+ 12)

(2πA)−sG(s− 1

2− iT ) ·

[ ∞∑n=1

τiT (n)τit(n)

ns

]ds (3.46)

We evaluate the Dirichlet series under the integral with Ramanujan’s identity (3.8)

∞∑n=1

τiT (n)τit(n)

ns=

∞∑n=1

σ2iT (n)σ2it(n)

ns+iT+it=

∏±,± ζ(s± iT ± it)

ζ(2s)

where the product is over all four possible choices of ± signs. The function G(s) has

already been computed at 3.45 :

G(s− 1

2− iT ) =

2s−3

s− 12− iT

·∏±,± Γ( s±it±iT

2)

Γ(s)

Hence the right-hand side of 3.46 equals

A12+iT

8· 1

2πi

∫

(2)

(πA)−s ·∏±,± Γ( s±it±iT

2)ζ(s± it± iT )

Γ(s)ζ(2s)· ds

s− 12− iT

Inserting the factor 16ξ(2τ)ξ(2s)

that was omitted for reasons of simplicity, we obtain the

following expression for T1 :

T1(T, t) =2A

12+iT

ξ(1 + 2it)ξ(1 + 2iT )· 1

2πi

∫

(2)

∏±,± ξ(s± it± iT )

ξ(2s)

A−sds

s− 12− iT

(3.47)

45

In view of our goal, which is the estimate 3.42 for the integral of |T1|2, we can assume,

without loss of generality, that t > 0 and t 6= T .

Returning to the identity 3.47, in order to obtain a useful estimate for T1 we need

to shift the line of integration from <s = 2 to <s = 12. This is possible primarily

because ζ(2s) 6= 0 on the line <s = 12. The reason why we cannot go even beyond this

line is that ζ(2s) has potential zeros close to the left of the line <s = 12

(we cannot

assume the existence of a zero-free region for the Riemann zeta).

However, there is still an obvious pole on the line <s = 12, namely at s = 1

2+ iT .

We are thus forced to have our line of integration (12) = <s = 1

2 make a small

loop to the left of s = 12

+ iT , so as to include this residue in the residue formula

(see below). Other poles that we encounter are the poles of the zeta factors from the

numerator of the integrand, on the line <s = 1.

We thus collect a total of five poles, namely:

s =1

2+ iT and s = 1± it± iT (3.48)

All of the above are simple poles (as t > 0 and t 6= T ). We denote by rj(T, t) the

corresponding residues, with 0 ≤ j ≤ 4. By the theorem of residues, we obtain the

identity

T1(T, t) =4∑

j=0

rj(T, t) + T ∗(T, t) (3.49)

where the term T ∗ represents the integral on the right-hand side of 3.47, shifted to

the line (12). Using the Cauchy-Schwartz inequality,

∫ ∞

0

|T1|2dt ≤ 6

(4∑

j=0

∫ ∞

0

|rj|2dt +

∫ ∞

0

|T ∗|2dt

)(3.50)

In what follows we shall find estimates for each of these integrals separately.

46

3.7.2 An estimate for the contribution of residues

The computation of the residues is straightforward. All we need to use is

Ress=1 ξ(s) = 1 (3.51)

and this follows from the identity ξ(s) = π−s2 Γ

(s2

)ζ(s) together with the well-known

facts: Γ(12) =

√π and Ress=1 ζ(s) = 1.

As a preliminary remark, we note that Stirling’s formula (3.14), together with the

estimate of ζ(s) on the line <s = 1 (3.16) guarantees, for an arbitrarily small ε > 0,

the existence of a constant M = M(ε) ≥ 1 such that the following inequality holds

for any real t:

1

M(1 + |t|)− 1

2+ε ≤

∣∣∣∣ξ(1 + 2it)

ξ(2 + 2it)

∣∣∣∣ = π12

∣∣∣∣Γ(1

2+ it)

Γ(1 + it)

ζ(1 + 2it)

ζ(2 + 2it)

∣∣∣∣ ≤ M(1 + |t|)− 12+ε

We denote by B(s) the integrand from eq. 3.47, whose residues at the poles from 3.48

we need to compute.

The term r1(T, t)

r1(T, t) := Res B, s = 1 + it + iT

=2A− 1

2−it

12

+ it· ξ(1 + 2it + 2iT )

ξ(2 + 2it + 2iT )=

2A− 12−it

12

+ it·O(

(1 + |t + T |)− 12+ε

)

for an arbitrarily small ε > 0. Hence

∫ ∞

0

|r1(T, t)|2dt ¿ε A−1

∫ ∞

0

(1 + |t + T |)−1+ε

1 + |t|2 dt ¿ε A−1T−1+ε (3.52)

47

The term r2(T, t)

r2(T, t) := Res B, s = 1− it + iT

=2A− 1

2+it

12− it

· ξ(1− 2it)

ξ(1 + 2it)· ξ(1− 2it + 2iT )

ξ(2− 2it + 2iT )=

2A− 12+it

12− it

·O((1 + |t− T |)− 1

2+ε

)


∫ ∞

0

|r2(T, t)|2dt ¿ε A−1

∫ ∞

0

(1 + |t− T |)−1+ε

1 + |t|2 dt ¿ε A−1T−1+ε (3.53)

The term r3(T, t)

r3(T, t) := Res B, s = 1 + it− iT

=2A− 1

2−it+2iT

12

+ i(t− 2T )· ξ(1− 2iT )

ξ(1 + 2iT )· ξ(1 + 2it + 2iT )

ξ(2 + 2it + 2iT )

=2A− 1

2−it+2iT

12

+ i(t− 2T )·O(

(1 + |t + T |)− 12+ε

)


∫ ∞

0

|r3(T, t)|2dt ¿ε A−1

∫ ∞

0

(1 + |t + T |)−1+ε

1 + |t− 2T |2 dt ¿ε A−1T−1+ε (3.54)

The term r4(T, t)

r4(T, t) := Res B, s = 1− it− iT

=2A− 1

2+it+2iT

12− i(t + 2T )

· ξ(1− 2it)

ξ(1 + 2it)· ξ(1− 2iT )

ξ(1 + 2iT )· ξ(1− 2it− 2iT )

ξ(2− 2it− 2iT )

=2A− 1

2+it+2iT

12− i(t + 2T )

·O((1 + |t + T |)− 1

2+ε

)

48


∫ ∞

0

|r4(T, t)|2dt ¿ε A−1

∫ ∞

0

(1 + |t + T |)−1+ε

1 + |t + 2T |2 dt ¿ A−1T−2 (3.55)

The term r0(T, t)

r0(T, t) := ResB, s =1

2+ iT

=2|ξ(1

2+ it)|2

ξ(1 + 2it)ξ2(1 + 2iT )·∏±

ξ(1

2+ 2iT ± it)

= π−12+it · |Γ(1

4+ it/2)|2 ∏

± Γ(14

+ iT ± it/2)

Γ(12

+ it)Γ2(12

+ iT )· |ζ(1

2+ it)|2 ∏

± ζ(12

+ 2iT ± it)

ζ(1 + 2it)ζ2(1 + 2iT )

Stirling’s formula and the estimate 3.16 yield once again

|r0(T, t)| ¿T ε(1 + |t|)ε

(1 + |t|)− 1

2 (1 + |t + 2T |)− 14 (1 + |t− 2T |)− 1

4

× |ζ(1

2+ it)|2

∏±|ζ(

1

2+ 2iT ± it)| · exp

π

4

(2T − t− |t− 2T |)

Using this bound, we split the integral of |r0|2 as∫∞

0=

∫ 4T

0+

∫∞4T

and obtain (up to

T ε)

∫ ∞

0

|r0(T, t)|2dt ¿∫ 4T

0

|ζ(12

+ it)|41 + |t|

∏±

|ζ(12

+ i(t± 2T ))|2(1 + |t± 2T |)1/2

dt

+

∫ ∞

4T

|ζ(12

+ it)|41 + |t|

∏±

|ζ(12

+ i(t± 2T ))|2(1 + |t± 2T |)1/2

· exp(− π

2(t− 2T )

)dt

(3.56)

For 0 ≤ t ≤ 4T , the subconvexity estimate for ζ(s) on the critical line (3.18) yields

∏±

|ζ(12

+ i(t± 2T ))|2(1 + |t± 2T |)1/2

¿ |ζ(12

+ i(t + 2T ))|2(1 + |t + 2T |)1/2

¿ T−1/2+2θ

49

In view of the fourth moment of the Riemann zeta (3.24), the first integral of the

right-hand side of 3.56 is then bounded by

T−1/2+2θ ·∫ 4T

0

|ζ(12

+ it)|41 + |t| dt ¿ T−1/2+2θ log5 T ¿ T−1/6,

since θ < 16

(3.20).

The second integral on the right-hand side of 3.56 is a negligible quantity, since

in this range the exponential decay takes over; this integral is trivially bounded by:

∫ ∞

4T

exp(− π

2(t− 2T )

)dt =

2

πe−πT

We conclude that ∫ ∞

0

|r0(T, t)|2dt = O(T−1/6) (3.57)

with the implied constant absolute.

We collect the results of this section (eq. 3.52, 3.53, 3.54, 3.55, and 3.57) in the

next proposition.

Proposition 3.7. There exists an absolute constant B3 such that the following in-

equality4∑

j=0

∫ ∞

0

|rj(T, t)|2dt ≤ B3T−1/6

holds for T ≥ 1.

3.8 An Estimate for∫∞

0 |T ∗(T, t)|2dt

In this section we prove that

∫ ∞

0

|T ∗(T, t)|2dt = O(T−1/6) (3.58)

50

In view of 3.50, this would finish the proof of 3.42 and, implicitly, that of Proposition

3.6. We return to this sequence of implications at the end of the chapter.

Remark 3.5. As was the case in the previous section, the exponent depends on the

strength of the subconvexity estimate 3.18. Essentially, the right-hand side of 3.58

is O(T−b), with b = 12− 2θ and θ the subconvex exponent from 3.18. Hence any

subconvex estimate (θ < 14) suffices for the purpose of showing that

∫ |T (T, t)|2dt has

polynomial decay as T →∞. For convenience, we will use θ < 16

(3.20).

A preliminary estimate

Recall the definition of T ∗ (eq. 3.49) :

T ∗(T, t) =2A

12+iT

ξ(1 + 2it)ξ(1 + 2iT )· 1

2πi

∫

(1/2)

∏±,± ξ(s± it± iT )

ξ(2s)

A−sds

s− 12− iT

Using Stirling’s asymptotic formula (3.14) for the gamma function built in the defi-

nition of ξ, we obtain a preliminary bound for T ∗. The next inequality holds up to

factors of size O(T ε(1 + |t|)ε

), for an arbitrarily small ε > 0. Such terms arise from

estimating the Riemann zeta on the line <s = 1, and are harmless since we eventually

prove a power saving. We have:

T ∗(T, t) ¿∫ ∞

−∞

∏±,±

|ζ(12

+ ix± it± iT )|(1 + |x± t± T |)1/4

· exp Ω(x, t, T )

1 + |x− T | dx (3.59)

The exponential factor Ω has the following expression

Ω(x, t, T ) =π

2

(|x|+ |t|+ |T |)− π

4

∑±,±

|x± t± T |, (3.60)

where the double sum is performed over all four combinations of ± signs. Since the

factor π4

has no significance in the analysis, we shall discard it and therefore consider

51

the exponential factor Ω to be given by

Ω(x, t, T ) = 2(|x|+ |t|+ |T |)−∑±,±

|x± t± T | (3.61)

By the triangle inequality, Ω ≤ 0. It is therefore legitimate for our purposes to ignore

the exponential factor whenever we find that convenient.

Starting with 3.59, we split the integral into three parts as follows:

T ∗(T, t) ¿∫ 0

−∞dx +

∫ 4T

0

dx +

∫ ∞

4T

dx =: g−1(t) + g1(t) + g2(t) (3.62)

We take 3.62 as the definition of g−1, g1 and g2. It is evident from the explicit form of

the integrand (3.59) that g−1(t) ≤ g1(t) + g2(t). Therefore, by the Cauchy-Schwartz

inequality, |T ∗(T, t)|2 ≤ 9(g1(t)

2 + g2(t)2)

and hence

∫ ∞

0

|T ∗(T, t)|2dt ≤ 9(∫ ∞

0

|g1(t)|2dt +

∫ ∞

0

|g2(t)|2dt)

(3.63)

It is thus enough to estimate the integrals∫∞

0|g1(t)|2dt and

∫∞0|g2(t)|2dt. For conve-

nience, we further split these integrals after the intervals 0 ≤ t ≤ 4T and 4T ≤ t < ∞,

and analyze each range separately.

The way we proceed in finding appropriate estimates is roughly the following:

each term gi is defined by an integral expression which contains four zeta factors; we

apply the subconvexity estimate for two of these factors, while we bring in the fourth

moment in order to bound the contribution of the remaining two.


0 |g1(t)|2dt

This is the range 0 ≤ t ≤ 4T and 0 ≤ x ≤ 4T . Here we choose to ignore the

exponential factor exp(Ω) since, as mentioned before, Ω ≤ 0. The first step is to

apply the mean-value theorem to the integral defining g1(t) (the integrand is positive)

52

to obtain

g1(t) =

∫ 4T

0

∏±,±

|ζ(12

+ i(x± T ± t))|(1 + |x± T ± t|)1/4

· dx

1 + |x− T | (3.64)

=∏±

|ζ(12

+ i(x0 ± T − t))|(1 + |x0 ± T − t|)1/4

·∫ 4T

0

∏±

|ζ(12

+ i(x± T + t))|(1 + |x± T + t|)1/4

· dx

1 + |x− T | ,

for some x0 in the interval [0, 4T ].

We now apply the subconvex estimate 3.18 to the first two zeta factors to obtain

∏±

|ζ(12

+ i(x0 ± T − t))|(1 + |x0 ± T − t|)1/4

¿ (1 + |x0 + T − t|)− 14+θ(1 + |x0 − T − t|)− 1

4+θ

Of the two factors 1 + |x0 + T − t| and 1 + |x0 − T − t|, at least one is greater than

T . Since θ < 16, the right-hand side is therefore bounded by T− 1

4+θ. Hence

g1(t) ¿ T− 14+θ ·

∫ 4T

0

∏±

|ζ(12

+ i(x± T + t))|(1 + |x± T + t|)1/4

· dx

1 + |x− T | (3.65)

Holder’s inequality ∣∣∣∣∫

X

fgdµ

∣∣∣∣2

≤ µ(X)‖f‖24‖g‖2

4 (3.66)

combined with 3.65 yields

|g1(t)|2 ¿ T− 12+2θ(log T ) ·

∫ 4T

0

|ζ(12

+ i(x + T + t))|41 + |x + T + t|

dx

1 + |x− T | 1

2

×∫ 4T

0

|ζ(12

+ i(x− T + t))|41 + |x− T + t|

dx

1 + |x− T | 1

2

(3.67)

The logarithmic factor comes from∫ 4T

0dx

1+|x−T | ¿ log T . It plays no role however,

since it will be absorbed in the power saving.

53

By the Cauchy-Schwartz inequality, applied to the integral of |g1|2, we have

∫ 4T

0

|g1(t)|2dt ¿ T− 12+2θ(log T )(J1J2)

1/2, (3.68)

with J1 and J2 defined by the following equations:

J1 =

∫ 4T

0

∫ 4T

0

|ζ(12

+ i(x + T + t))|41 + |x + T + t|

dxdt

1 + |x− T | (3.69)

and

J2 =

∫ 4T

0

∫ 4T

0

|ζ(12

+ i(x− T + t))|41 + |x− T + t|

dxdt

1 + |x− T | (3.70)

With the change of variable x− T = u, x + T + t = w, we rewrite J1 as

J1 =

∫ 3T

−T

∫ u+6T

u+2T

|ζ(12

+ iw)|41 + |w|

dwdu

1 + |u|

In view of the fourth moment of the Riemann zeta (3.24), the w-integral is bounded,

up to an absolute constant, by log5 T . Hence

J1 ¿ log5 T

∫ 3T

−T

du

1 + |u| ¿ log6 T (3.71)

The same argument works in the case of J2:

J2 =

∫ 3T

−T

∫ u

u−4T

|ζ(12

+ iw)|41 + |w|

du

1 + |u| ¿ log6 T (3.72)

Combining the estimates for J1 and J2 with eq. 3.68, we obtain

∫ 4T

0

|g1(t)|2dt ¿ε T− 12+2θ+ε, ∀ε > 0 (3.73)

54

3.8.2 The integral∫∞

4T |g1(t)|2dt

This is the range 4T ≤ t < ∞, 0 ≤ x ≤ 4T . In this range, the exponential factor

exp(Ω) plays a more significant role. We begin with the mean-value theorem applied

to the integral expression of g1(t). We separate the zeta factors as follows:

g1(t) =

∫ 4T

0

∏±,±

|ζ(12

+ i(x± T ± t))|(1 + |x± T ± t|)1/4

· exp(Ω)dx

1 + |x− T | (3.74)

=∏±

|ζ(12

+ i(x0 + T ± t))|(1 + |x0 + T ± t|)1/4

·∫ 4T

0

∏±

|ζ(12

+ i(x− T ± t))|(1 + |x− T ± t|)1/4

· exp(Ω)dx

1 + |x− T | ,

for some x0 in the interval [0, 4T ].

The subconvexity estimate (3.18) applied to the first two zeta factors yields

∏±

|ζ(12

+ i(x0 + T ± t))|(1 + |x0 + T ± t|)1/4

¿∏±

(1 + |x0 + T ± t|)− 14+θ ¿ t−

14+θ

hence

|g1(t)| ¿ t−14+θ

∫ 4T

0

∏±

|ζ(12

+ i(x− T ± t))|(1 + |x− T ± t|)1/4

· exp(Ω)dx

1 + |x− T | (3.75)

By Holder’s inequality (3.66) applied on the right-hand side,

|g1(t)|2 ¿ t−12+2θ(log T ) ·

∫ 4T

0

|ζ(12

+ i(x− T + t))|41 + |x− T + t|

exp(Ω)dx

1 + |x− T | 1

2

×∫ 4T

0

|ζ(12

+ i(x− T − t))|41 + |x− T − t|

exp(Ω)dx

1 + |x− T | 1

2

(3.76)

Here the logarithmic factor comes from

∫ 4T

0

exp(Ω)dx

1 + |x− T | ≤∫ 4T

0

dx

1 + |x− T | ¿ log T

55

Applying the Cauchy-Schwartz inequality to the right-hand side of 3.76, we have

∫ ∞

4T

|g1(t)|2dt ¿ (J1J2)1/2 log T, (3.77)

where J1, J2 are this time given by

J1 =

∫ ∞

4T

∫ 4T

0

t−12+2θ · |ζ(1

2+ i(x− T + t))|4

1 + |x− T + t|exp(Ω)dxdt

1 + |x− T | (3.78)

and

J2 =

∫ ∞

4T

∫ 4T

0

t−12+2θ · |ζ(1

2+ i(x− T − t))|4

1 + |x− T − t|exp(Ω)dxdt

1 + |x− T | (3.79)

To estimate J1, we first make the change of variable x − T = u, t + x − T = w, to

obtain

J1 =

∫ 3T

−T

∫ ∞

u+4T

|ζ(12

+ iw)|41 + |w|

|w − u|− 12+2θ exp(Ω)

1 + |u| dwdu

The exponent Ω is given in the new coordinates by

Ω = 2T − |2u− w| − |2u− w + 2T | (3.80)

In the range in question, |w− u|− 12+2θ ¿ T− 1

2+2θ (recall that θ < 1

6) and u + 4T ≥ T .

Therefore

J1 ¿ T− 12+2θ

∫ 3T

−T

∫ ∞

T

|ζ(12

+ iw)|41 + |w| exp(Ω)dw

du

1 + |u|

Once again we use the fourth moment (3.24) and the convexity bound (3.21) for ζ(s)

to estimate the inner integral:

∫ ∞

T

|ζ(12

+ iw)|41 + |w| exp(Ω)dw ¿

∫ 10T

T

|ζ(12

+ iw)|41 + |w| dw +

∫ ∞

10T

exp(Ω)dw

¿ log4 T +

∫ ∞

10T

exp(4T + 4u− 2w)dw ¿ log4 T,

56

since u ≤ 4T . Therefore

J1 ¿ T− 12+2θ(log4 T ) ·

∫ 3T

−T

du

1 + |u| ¿ T− 12+2θ log5 T (3.81)

Similarly in the case of J2, the change of variable x− T = u, t + T − x = w yields

J2 =

∫ 3T

−T

∫ ∞

4T−u

|ζ(12

+ iw)|41 + |w|

|u + w|− 12+2θ exp(Ω)

1 + |u| dwdu

In the new coordinates, the exponent Ω is given by

Ω = 2T − |w| − |w − 2T | (3.82)

The integration variables satisfy u + w ≥ 4T and hence |u + w|− 12+2θ ¿ T− 1

2+2θ;

because w ≥ 4T − u ≥ T and the integrand is positive,

J2 ¿ T− 12+2θ

∫ 3T

−T

∫ ∞

T

|ζ(12

+ iw)|41 + |w| exp(Ω)dw

du

1 + |u|

In the same way as before,

∫ ∞

T

|ζ(12

+ iw)|41 + |w| exp(Ω)dw ¿ log4 T

hence

J2 ¿ T− 12+2θ log5 T (3.83)

Combining the estimates for J1 and J2 with eq. 3.77, we are led to

∫ ∞

4T

|g1(t)|2dt ¿ε T− 12+2θ+ε, ∀ε > 0 (3.84)

57


0 |g2(t)|2dt

This is the range 0 ≤ t ≤ 4T, 4T ≤ x < ∞. Even if the integral defining g2 is in this

case over the non-compact interval [4T,∞), we can still apply the mean-value theo-

rem, since the integrand is a product φ(x)ψ(x), with ψ(x) = o(x), x → ∞. Once we

have that, we obtain, by the mean value theorem,∫∞4T

φ(x)ψ(x)dx = ψ(x0)∫∞4T

φ(x)dx,

for some x0 in the interval [4T,∞). Here ψ(x) =∏±|ζ( 1

2+i(t+T±x))|

(1+|t+T±x|)1/4 = o(x−12+2θ), while

φ(x) =∏±|ζ( 1

2+i(t−T±x))|

(1+|t−T±x|)1/4 · exp(Ω)1+|x−T | . Therefore we obtain:

g2(t) =

∫ ∞

4T

∏±,±

|ζ(12

+ i(t± T ± x))|(1 + |t± T ± x|)1/4

· exp(Ω)dx

1 + |x− T | (3.85)

=∏±

|ζ(12

+ i(t + T ± x0))|(1 + |t + T ± x0|)1/4

·∫ ∞

4T

∏±

|ζ(12

+ i(t− T ± x))|(1 + |t− T ± x|)1/4

· exp(Ω)dx

1 + |x− T | ,

for some x0 in the interval [4T,∞).

We apply the subconvexity estimate to the zeta factor containing +x0, while to

the other zeta factor in front of the integral we apply the convexity estimate:

∏±

|ζ(12

+ i(t + T ± x0))|(1 + |t + T ± x0|)1/4

¿ |t + T + x0|− 14+θ ≤ T− 1

4+θ

Therefore

g2(t) ¿ T− 14+θ

∫ ∞

4T

∏±

|ζ(12

+ i(t− T ± x))|(1 + |t− T ± x|)1/4

· exp(Ω)dx

1 + |x− T |

By applying Holder’s inequality (3.66) as before, we obtain

|g2(t)|2 ¿ T− 12+2θ

∫ ∞

4T

exp(Ω)dx

1 + |x− T |∫ ∞

4T

|ζ(12

+ i(t− T + x))|41 + |t− T + x|

exp(Ω)dx

1 + |x− T | 1

2

×∫ ∞

4T

|ζ(12

+ i(t− T − x))|41 + |t− T − x|

exp(Ω)dx

1 + |x− T | 1

2

(3.86)

58

The first integral on the right-hand side has logarithmic growth

∫ ∞

4T

exp(Ω)dx

1 + |x− T | ≤∫ 10T

4T

dx

1 + |x− T | +

∫ ∞

10T

exp 2(t + T − x)dx ¿ log T,

since in the current case t is confined to the interval [4T,∞), and hence the second

integral has exponential decay in T .

Combined with the Cauchy-Schwartz inequality, eq. 3.86 yields a bound for the

integral of |g2|2 itself

∫ 4T

0

|g2(t)|2dt ¿ T− 12+2θ(log T )(J1J2)

1/2 (3.87)

where J1, J2 are the double integrals:

J1 =

∫ 4T

0

∫ ∞

4T

|ζ(12

+ i(t− T + x))|41 + |t− T + x|

exp(Ω)dxdt

1 + |x− T | (3.88)

and

J2 =

∫ 4T

0

∫ ∞

4T

|ζ(12

+ i(t− T − x))|41 + |t− T − x|

exp(Ω)dxdt

1 + |x− T | (3.89)

The change of variable x− T = u, t + x− T = w in the expression for J1 yields

J1 =

∫ ∞

3T

∫ u+4T

u

|ζ(12

+ iw)|41 + |w|

exp(Ω)dwdu

1 + |u|

with the exponential factor in the new coordinates

Ω = 2T − |2u− w| − |2u− w + 2T | (3.90)

59

Therefore

J1 ≤∫ 4T

3T

∫ u+4T

u

|ζ(12

+ iw)|41 + |w| dw

du

1 + |u| +

∫ ∞

4T

∫ u+4T

u

|ζ(12

+ iw)|41 + |w|

e−|u−4T |

1 + udwdu

¿ log4 T

∫ 4T

3T

du

1 + u+

∫ ∞

4T

e−|u−4T |

1 + ulog4 udu

¿ log4 T + T−1 log4 T ¿ log4 T (3.91)

Similarly, in the case of J2, the change of variable x− T = u, x + T − t = w yields

J2 =

∫ ∞

3T

∫ u+2T

u−2T

|ζ(12

+ iw)|41 + |w|

exp(Ω)

1 + |u| dwdu

≤∫ 4T

3T

∫ u+2T

u−2T

|ζ(12

+ iw)|41 + |w| dw

du

1 + |u| +

∫ ∞

4T

∫ u+2T

u−2T

|ζ(12

+ iw)|41 + |w|

e4T−2w

1 + udwdu

¿∫ 4T

3T

∫ 6T

T

|ζ(12

+ iw)|41 + |w| dw

du

1 + u+

∫ ∞

4T

∫ u+2T

u−2T

|ζ(12

+ iw)|41 + |w| dw

e6T−2u

1 + udu

¿ log4 T

∫ 4T

3T

du

1 + u+

∫ ∞

4T

(log u)4 exp(8T − 2u)du

1 + u

¿ log4 T (3.92)

Combining the estimates for J1, J2 with eq. 3.87, we obtain

∫ 4T

0

|g2(t)|2dt ¿ε T− 12+2θ+ε, ∀ε > 0 (3.93)

3.8.4 The integral∫∞

4T |g2(t)|2dt

This is the case t ≥ 4T, x ≥ 4T . We apply the mean-value theorem as before:

g2(t) =

∫ ∞

4T

∏±,±

|ζ(12

+ i(x± t± T ))|(1 + |x± t± T |)1/4

· exp(Ω)dx

1 + |x− T | (3.94)

=∏±

|ζ(12

+ i(x0 + t± T ))|(1 + |x0 + t± T |)1/4

·∫ ∞

4T

∏±

|ζ(12

+ i(x− t± T ))|(1 + |x− t± T |)1/4

· exp(Ω)dx

1 + |x− T | ,

60

for some x0 ∈ [4T,∞).

We apply the subconvexity estimate (3.18) to the first zeta factor outside the

integral; for the second ratio outside the integral we only need to know that it is

bounded by an absolute constant, in view of 3.21. Therefore

∏±

|ζ(12

+ i(x0 + t± T ))|(1 + |x0 + t± T |)1/4

¿ (1 + |x0 + t + T |)− 14+θ ≤ t−

14+θ

and hence

g2(t) ¿ t−14+θ

∫ ∞

4T

∏±

|ζ(12

+ i(x− t± T ))|(1 + |x− t± T |)1/4

· exp(Ω)dx

1 + |x− T | (3.95)

By Holder’s inequality (3.66) once again,

|g2(t)|2 ¿ t−12+2θ

∫ ∞

4T

exp(Ω)dx

1 + |x− T |∫ ∞

4T

|ζ(12

+ i(x− t + T ))|41 + |x− t + T |

exp(Ω)dx

1 + |x− T | 1

2

×∫ ∞

4T

|ζ(12

+ i(x− t− T ))|41 + |x− t− T |

exp(Ω)dx

1 + |x− T | 1

2

(3.96)

Just as before, the first integral is bounded by log t, therefore it can be ignored. By

applying the Cauchy-Schwartz inequality once more, we obtain a preliminary bound

for the integral of |g2|2 ∫ ∞

4T

|g2(t)|2dt ¿ (J1J2)1/2 (3.97)

where the quantities J1, J2 are in this case

J1 =

∫ ∞

4T

∫ ∞

4T

t−12+2θ · |ζ(1

2+ i(x− t + T ))|4

1 + |x− t + T |exp(Ω)dxdt

1 + |x− T | (3.98)

and

J2 =

∫ ∞

4T

∫ ∞

4T

t−12+2θ · |ζ(1

2+ i(x− t− T ))|4

1 + |x− t− T |exp(Ω)dxdt

1 + |x− T | (3.99)

61

The term J1

The change of variable x− T = u, t− x− T = w in the expression for J1 yields

J1 =

∫ ∞

3T

∫ ∞

2T−u

|ζ(12

+ iw)|41 + |w|

|u + w + 2T |− 12+2θ exp(Ω)

1 + |u| dwdu

The exponent Ω is given in the new coordinates by

Ω = 2T − |w| − |w + 2T | (3.100)

In order to estimate J1, we split this integral in three parts as follows:

J1 =

∫ ∞

3T

∫ 0

2T−u

dwdu +

∫ ∞

3T

∫ u

0

dwdu +

∫ ∞

3T

∫ ∞

u

dwdu

=: J11 + J12 + J13 (3.101)

With a further change of variable w 7→ −w we have

J11 =

∫ ∞

3T

∫ u−2T

0

|ζ(12

+ iw)|41 + |w|

|u− w + 2T |− 12+2θ exp(Ω)

1 + |u| dwdu (3.102)

and the exponent is now Ω = 2T − |w| − |w− 2T |. When w is a fraction of u, we will

use the fact that |u − w + 2T | À u and hence |u − w + 2T |− 12+2θ ¿ u−

12+2θ. When

u is large and w is larger than a fraction of u, the exponent Ω is large in absolute

value, and the exponential decay takes over. This can be expressed by breaking up

62

the integral into appropriate ranges as follows:

J11 =

∫ 6T

3T

∫ u−2T

0

dwdu +

∫ ∞

6T

∫ u2

0

dwdu +

∫ ∞

6T

∫ u−2T

u2

dwdu

¿ T− 12+2θ

∫ 6T

3T

∫ u−2T

0

|ζ(12

+ iw)|41 + |w|

dwdu

1 + |u| +

∫ ∞

6T

∫ u2

0

|ζ(12

+ iw)|41 + |w| u−

12+2θ dwdu

1 + |u|

+T− 12+2θ

∫ ∞

6T

∫ u−2T

u2

|ζ(12

+ iw)|41 + |w|

exp(4T − u)dwdu

1 + |u|

¿ T− 12+2θ

∫ 6T

3T

(log u)5 du

1 + |u| +

∫ ∞

6T

u−12+2θ(log u)5 du

1 + |u|+T− 1

2+2θ

∫ ∞

6T

(log u)5 exp(4T − u)

1 + |u| du (3.103)

hence

J11 ¿ T− 12+2θ log5 T (3.104)

The other terms of the equation 3.101, J12 and J13, admit a simpler treatment:

J12 =

∫ ∞

3T

∫ u

0

|ζ(12

+ iw)|41 + |w|

|u + w + 2T |− 12+2θ exp(−2w)

1 + udwdu

¿∫ ∞

3T

∫ u

0

|ζ(12

+ iw)|41 + |w| dw

u−12+2θdu

1 + u

¿∫ ∞

3T

u−12+2θ(log u)5 du

1 + u¿ T− 1

2+2θ log5 T (3.105)

and

J13 =

∫ ∞

3T

∫ ∞

u

|ζ(12

+ iw)|41 + |w|

|u + w + 2T |− 12+2θ exp(−2w)

1 + udwdu

¿ T− 12+2θ

∫ ∞

3T

u−12+2θ exp(−2u)

du

1 + u

¿ exp(−6T ) = (negligible) (3.106)

Collecting the estimates for J11, J12 and J13 (3.104, 3.105, 3.106) we obtain an estimate

for the term J1 of 3.97

J1 ¿ T− 12+2θ log5 T (3.107)

63

The term J2

For the term J2 of 3.97, we make the substitution x− T = u, t + T − x = w

J2 =

∫ ∞

3T

∫ ∞

4T−u

|ζ(12

+ iw)|41 + |w|

|u + w|− 12+2θ exp(Ω)

1 + udwdu (3.108)

with the exponential factor now given by

Ω = 2T − |w| − |w − 2T |. (3.109)

We split the last integral

J2 =

∫ 10T

3T

∫ ∞

4T−u

dwdu +

∫ ∞

10T

∫ −u2

4T−u

dwdu +

∫ ∞

10T

∫ ∞

−u2

dwdu

=: J21 + J22 + J23 (3.110)

The quantity J21 is bounded by

T− 12+2θ

∫ 10T

3T

∫ ∞

4T−u

|ζ(12

+ iw)|41 + |w| exp(Ω)dw

du

1 + u

The inner integral is less than

∫ u

4T−u

|ζ(12

+ iw)|41 + |w| dw +

∫ ∞

u

|ζ(12

+ iw)|41 + |w| exp(4T − 2w)dw

¿∫ u

−u

|ζ(12

+ iw)|41 + |w| dw +

∫ ∞

u

exp(4T − 2w)dw ¿ log5 T.

We have

J21 ¿ T− 12+2θ log5 T

∫ 10T

3T

du

1 + u¿ T− 1

2+2θ log5 T (3.111)

64

In the case of J22, a further change of variable w 7→ −w yields

J22 =

∫ ∞

10T

∫ u−4T

u2

|ζ(12

+ iw)|41 + |w|

|u− w|− 12+2θ exp(Ω)

1 + udwdu

≤ T− 12+2θ

∫ ∞

10T

∫ u−4T

u2

|ζ(12

+ iw)|41 + |w| exp(−2w)dw

du

1 + u

¿ T− 12+2θ

∫ ∞

10T

exp(−u)du ¿ exp(−T ) = (negligible) (3.112)

and here we only used the fact that the function|ζ( 1

2+iw)|4

1+|w| is bounded.

In the case of J23, we have u + w ≥ u2

and hence |u + w|− 12+2θ ¿ u−

12+2θ, so that

J23 ¿∫ ∞

10T

∫ ∞

−u2

|ζ(12

+ iw)|41 + |w| exp(Ω)u−

12+2θdw

du

1 + u

Using the fourth moment of ζ(s) once again, the inner integral is bounded by

∫ 4u

−u2

|ζ(12

+ iw)|41 + |w| dw +

∫ ∞

4u

exp(4T − 2w)dw

¿ log5 u + exp(4T − 8u) ¿ log5 u

since in this range u > 4T . We obtain a bound for J23:

J23 ¿∫ ∞

10T

(log u)5u−12+2θ du

1 + u¿ T− 1

2+2θ log5 T (3.113)

Collecting the results obtained at 3.111, 3.112 and 3.113, we have

J2 ¿ T− 12+2θ log5 T (3.114)

Eq. 3.97 states that∫∞

4T|g2(t)|2dt ¿ (J1J2)

1/2. Using now the estimates for J1 and

J2 from 3.107 and 3.114, we infer that

∫ ∞

4T

|g2(t)|2dt ¿ε T− 12+2θ+ε, ∀ε > 0 (3.115)

65

3.9 Conclusion

Let us recall the inequality from 3.63:

∫ ∞

0

|T ∗(T, t)|2dt ≤ 9

(∫ ∞

0

|g1(t)|2dt +

∫ ∞

0

|g2(t)|2dt

)

The estimates obtained at 3.73,3.84, 3.93 and 3.115 on the integrals of |g1|2 and |g2|2

allow us to conclude that

∫ ∞

0

|T ∗(T, t)|2dt ¿ε T− 12+2θ+ε, ∀ε > 0 (3.116)

and θ is the exponent in the subconvexity estimate (3.18). The results of the equations

3.40, 3.49, 3.58, as well as proposition 3.7 are the following:

1.∫R |T (T, t)|2dt ≤ 8

∫∞0|T1(T, t)|2dt

2. T1(T, t) =∑4

j=0 rj(T, t) + T ∗(T, t)

3.∑4

j=0

∫∞0|rj(T, t)|2dt = O(T− 1

2+2θ+ε)

4.∫∞0|T ∗(T, t)|2dt = O(T− 1

2+2θ+ε)

This completes the proof of Proposition 3.6, namely

∫

R|T (T, t)|2dt = O(T−b), as T →∞.

with b = 12− 2θ + ε.

Remark 3.6. The condition θ < 16

yields b > 16.

By the inequality 3.13, we have a bound for the continuous spectrum contribution:

Cont.(A; T ) ≤ 3

4π· ∫

R|R(T, t)|2dt +

∫

R|M(T, t)|2dt +

∫

R|T (T, t)|2dt

66

The results from propositions 3.4, 3.5 and 3.6 are as follows

1.∫R |R(T, t)|2dt = O(T−1/6)

2.∫R |M(T, t)|2dt ≤ 144πA

3.∫R |T (T, t)|2dt = O(T−1/6)

This finishes the proof of Theorem 3.3:

Cont.(A; T ) ≤ 108A + O(T−1/6) (3.117)

with the implied constant absolute.

67

Chapter 4

Discrete Spectrum Contribution

4.1 Preliminary Remarks

Let us recall the spectral expansion of the L4 norm of EA(12

+ iT ):

∥∥EA(1

2+ iT )

∥∥4

4= Disc.(A; T ) + Cont.(A; T )

The term Disc.(A; T ) represents the contribution of the discrete spectrum and is given

explicitly by

Disc.(A; T ) :=∑

φ

∣∣∣⟨E2

A(1

2+ iT ), φ

⟩∣∣∣2

(4.1)

and it depends on the truncation parameter A (fixed) and on the spectral parameter

T → ∞. φ ranges over the entire countable family of L2-normalized Hecke-Maass

forms. The inner product is taken in L2(Γ\H).

As we have already seen in the previous chapter, the continuous spectrum contri-

bution is bounded, as T →∞. Therefore, in order to prove

∥∥EA(1

2+ iT )

∥∥L4(Γ\H)

= O(T ε), T →∞

68

we need to show that

Disc.(A; T ) = O(T ε), ∀ε > 0 (4.2)

and the implied constant should depend on ε and A only.

The proof of this estimate consists of two steps. First, we replace the quantity

Disc.(A; T ) by an arithmetic substitute Disc.(∞; T ) which depends only on T . To

justify this step we need to prove that

Disc.(A; T ) ¿ Disc.(∞; T ) + O(T ε) (4.3)

This is the content of Theorem 4.2 of the current chapter.

The second part consists of showing that Disc.(∞; T ) itself is O(T ε). As this is

a problem in the analytic theory of automorphic L-functions and has nothing to do

with L4 norms of eigenfunctions any longer, we devote a separate chapter to it. The

result concerning Disc.(∞; T ) will be formulated in Theorem 5.1 of Chapter 5.

4.2 An Arithmetic Substitute

The arithmetic substitute for Disc.(A; T ) is the quantity

Disc.(∞; T ) =∑

φ

∣∣∣⟨E2(

1

2+ iT ), φ

⟩∣∣∣2

(4.4)

The parameter A has been replaced by ∞ to suggest that there is no truncation

implicit in this quantity, which depends solely on the spectral parameter T . The

reason that each summand is well defined is that the Eisenstein series has polynomial

growth in the cusp, while a cusp form φ has exponential decay; hence the pairing⟨E2(1

2+ iT ), φ

⟩is well defined. It is not a priori obvious why the series defining

Disc.(∞; T ) should converge; this will become transparent during the proof.

The arithmetic nature of this quantity is a consequence of a triple product formula

69

due to Luo and Sarnak [L-S, eq. 17]. This is the subject of the next section.

4.2.1 The formula of Luo and Sarnak

Theorem 4.1 (Luo, Sarnak). Let φ be a Hecke-Maass cusp form and E(z, s) the

Eisenstein series on SL(2,Z). We have

∫

X

E2(z,1

2+ iT )φ(z)dz = w(tφ, 2T )L(

1

2, φ)L(

1

2+ 2iT, φ) (4.5)

with the weight w given by

w(tφ, 2T ) =ρφ(1)

2ζ2(1 + 2iT )· |Γ(1

4+ itφ/2)|2

Γ2(12

+ iT )·∏±

Γ(1

4+ i(T ± tφ/2))

Proof. For the sake of completeness, we give a proof of this identity which follows

word by word the proof in [L-S]. The argument is based on the ’unfolding’ principle

from the theory of the Rankin-Selberg integral.

Assume for the moment that the Maass form φ is fixed, as well as the parameter

T . For s ∈ C, the integral

I(s) :=

∫

Γ\HE(z,

1

2+ iT )E(z, s)φ(z)dz

is rapidly convergent, for the same, previously enumerated, reasons. Therefore I(s)

defines a meromorphic function in s ∈ C whose poles are determined by those of the

Eisenstein series. If φ is an odd cusp form, I(s) ≡ 0 by symmetry; on the other hand,

L(12, φ) = 0, and hence the identity of the theorem holds trivially in this case.

We will assume then that φ is an even cusp form, i.e.

ρφ(n) = ρφ(−n) = ρφ(1)λφ(n), n ≥ 1

70

If <s > 1 the Eisenstein series converges absolutely and, by unfolding the integral,

we obtain

I(s) =

∫

Γ\H

( ∑

γ∈Γ∞\Γys(γz)

)E(z,

1

2+ iT )φ(z)dz =

∫

Γ∞\HysE(z,

1

2+ iT )φ(z)dz

=

∫ ∞

0

∫ 1

0

ys−1

[e(y,

1

2+ iT ) +

2

ξ(1 + 2iT )

∑

n 6=0

τiT (n)√

yKiT (2π|n|y)e(nx)

]

×[∑

n 6=0

ρφ(n)√

yKitφ(2π|n|y)e(nx)

]dx

dy

y

=4ρφ(1)

ξ(1 + 2iT )

∞∑n=1

τiT (n)λφ(n)

(2πn)s·∫ ∞

0

ysKitφ(y)KiT (y)dy

y(4.6)

Employing once again the Mellin-Barnes formula (3.7)

∫ ∞

0

ysKitφ(y)KiT (y)dy

y= 2s−3

∏±,± Γ(

s±itφ±iT

2)

Γ(s)

and an identity of Ramanujan type (consequence of the Hecke relations 2.4)

∞∑n=1

τiT (n)λφ(n)

ns=

∞∑n=1

σ2iT (n)λφ(n)

ns+iT=

L(s + iT, φ)L(s− iT, φ)

ζ(2s)

we obtain

I(s) =ρφ(1)

2πsξ(1 + 2iT )·∏±,± Γ(

s±itφ±iT

2)

Γ(s)· L(s + iT, φ)L(s− iT, φ)

ζ(2s)(4.7)

So far we made the assumption that <s > 1. Since both sides of 4.7 define meromor-

phic functions in s, the identity must be valid for any s ∈ C. If we let s = 12

+ iT we

arrive at the desired formula.

71

Watson’s formula

In [Wa], T. Watson generalized the triple-product formula of Luo and Sarnak to

arbitrary Maass forms. In the case when Γ = SL(2,Z), this formula reads

∣∣∣∣∫

X

φ1(z)φ2(z)φ3(z)dz

∣∣∣∣2

=π4

216

Λ(12, φ1 ⊗ φ2 ⊗ φ3)∏3

j=1 Λ(1, sym2 φj)(4.8)

and in particular ∣∣∣⟨ψ2, φ

⟩∣∣∣2

=π4

216

Λ(12, sym2 ψ ⊗ φ)Λ(1

2, φ)∏3

j=1 Λ(1, sym2 φj)

In the special case ψ = E(z, 12

+ iT ), the above functorial tensor product L-function

splits

L(1

2, sym2 ψ ⊗ φ) = L(

1

2+ 2iT, φ)L(

1

2− 2iT, φ)L(

1

2, φ)

and we can see that the formula 4.8 implies the result of Theorem 4.1. However, the

proof of 4.8 is very involved, and the result is quite deep.

We also note that Watson’s formula, together with an analogue for GL3 of the

Voronoi summation formula developed by Miller and Schmid [M-S], represent the

main ingredients in the analysis of the L4 norm of a Maass cusp form (see [Sa-Wa]).

4.2.2 Controlling the difference

In this section we prove that the arithmetic quantity Disc.(∞; T ) is an admissible

substitute for Disc.(A; T ), the discrete spectrum contribution.

Theorem 4.2. Suppose T →∞. Then

∣∣∣Disc.(A; T )12 −Disc.(∞; T )

12

∣∣∣ = O(log(AT )) + O(A1/4T−1/12)

72

and the implied constants are absolute. Therefore, when A is fixed,

Disc.(A; T ) ≤ 2 Disc.(∞; T ) + O(log2 T )

Unless specified otherwise, in this section we will write EA instead of EA(z, 12+iT ),

and E instead of E(z, 12

+ iT ). Suppose φ is a Hecke-Maass cusp form. Since the

functions E and EA agree on FA but not on CA = z ∈ F∣∣y(z) ≥ A, where they

differ by the constant term, we have

⟨E2, φ

⟩− ⟨E2

A, φ⟩

=

∫

CA

(e2(y,

1

2+ iT ) + 2e(y,

1

2+ iT )EA(z)

)φ(z)dz

=

∫

CA

2e(y,1

2+ iT )EA(z)φ(z)dz =

∫

FH(z)φ(z)dz = 〈H, φ〉 (4.9)

where H(z) is the rapidly decreasing function defined by

H(z) =

0, z ∈ FA

2e(y, 12

+ iT )EA(z, 12

+ iT ), z ∈ CA

(4.10)

(Recall that CA is ’the cusp’ : z ∈ F∣∣y(z) ≥ A.)

Equation 4.9 shows that H and E2 − E2A have the same projection onto L2

0(X), the

space of cusp forms of X. If we denote by ‖ · ‖d the L2 norm restricted to L20 we

obtain, by the triangle inequality,

∣∣‖E2A‖d − ‖E2‖d

∣∣ ≤ ‖H‖d

Note that ‖E2A‖2

d = Disc.(A; T ), while ‖E2‖2d = Disc.(∞; T ). Since ‖H‖d ≤ ‖H‖2

(this is the content of Bessel’s inequality), we obtain

∣∣∣Disc.(A; T )12 −Disc.(∞; T )

12

∣∣∣ ≤ ‖H‖2 (4.11)

73

In this way we reduce the problem of controlling the difference between the discrete

spectrum contribution and the arithmetic substitute to that of finding an estimate

for ‖H‖2. This will be the subject of the next section.

4.3 An Estimate for ‖H‖2

In the remaining section of this chapter we prove the following estimate:

Proposition 4.3. As T →∞,

‖H‖2 = O(log(AT )

)+ O

(A1/4T−1/12

),

and the implied constants are absolute.

The computations done in this section are very similar to those of Section 2.3,

where we determined the L2 norm of an Eisenstein series. From the definition of H,

we have

‖H‖22 =

∫

X

|H(z)|2dz =

∫

CA

4∣∣e(y,

1

2+ iT )

∣∣2∣∣EA(z,1

2+ iT )

∣∣2dz (4.12)

The constant term is given explicitly by e(y, 12

+ iT ) = y12+iT + φ(1

2+ iT )y

12−iT , with

|φ(12

+ iT )| = 1. Hence |e(y, 12

+ iT )| ≤ 2y12 . Using the explicit Fourier expansion of

the Eisenstein series (2.7) we obtain

∫

X

|H(z)|2dz ≤ 16

∫

CA

y∣∣EA(z +

1

2+ iT )

∣∣2dz

=128

|ξ(1 + 2iT )|2∞∑

n=1

τ 2iT (n)

∫ ∞

A

K2iT (2πny)dy (4.13)

74

Mellin transform

We use the Mellin transform in order to evaluate the sum from 4.13. We denote by

g(x) the function

g(x) :=

∫ ∞

x

K2iT (y)dy (4.14)

Hence g(x) is rapidly decreasing at infinity. Let G(s) be the Mellin transform

G(s) :=

∫ ∞

0

g(x)xs dx

x(4.15)

By partial integration, G(s) = −1s

∫∞0

g′(x)xsdx = 1s

∫∞0

K2iT (x)xsdx, and by the

Mellin-Barnes formula we have

G(s) =2s−2

s

Γ2( s+12

)Γ( s+12

+ iT )Γ( s+12− iT )

Γ(s + 1)

G(s) is holomorphic when Re s > 0 and the Mellin inversion formula allows us to

express g(x) in terms of G(s) : g(x) = 12πi

∫(2)

G(s)x−sds. The notation means that

we integrate over the vertical line <s = 2. We can now evaluate the right-hand side

of 4.13:

∞∑n=1

τ 2iT (n)

2πn·∫ ∞

2πnA

K2iT (y)dy =

∞∑n=1

τ 2iT (n)

2πng(2πnA)

=∞∑

n=1

τ 2iT (n)

2πn· 1

2πi

∫

(2)

G(s)(2πnA)−sds

=2−3π−1

2πi

∫

(2)

(Aπ)−s · Γ2( s+12

)∏± Γ( s+1

2± iT )

Γ(s + 1)·[ ∞∑

n=1

τ 2iT (n)

ns+1

]ds

s

=2−3π−1

2πi

∫

(3)

(Aπ)1−s · Γ2( s2)∏± Γ( s

2± iT )

Γ(s)·[ ∞∑

n=1

τ 2iT (n)

ns

]ds

s− 1(4.16)

75

Ramanujan’s identity (3.8) yields

∞∑n=1

τ 2iT (n)

ns=

∞∑n=1

σ22iT (n)

ns+2iT=

ζ2(s)ζ(s + 2iT )ζ(s− 2iT )

ζ(2s)

Therefore, the right-hand side of 4.16 equals

2−3A

2πi

∫

(3)

(Aπ)−s · Γ2( s2)ζ2(s)

Γ(s)ζ(2s)

∏±

Γ(s

2± iT )ζ(s± 2iT )

ds

s− 1

=2−3

2πi

∫

(σ)

ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)

A1−sds

s− 1

Inserting the factor 128|ξ(1+2iT )|2 from 4.13 we obtain

‖H‖22 ≤

16

|ξ(1 + 2iT )|2 ·1

2πi

∫

(3)

ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)

A1−sds

s− 1(4.17)

We denote by B(s) the integrand from the last equation. In order to obtain a useful

estimate for the integral, we need to shift the line of integration from <s = 3 to

<s = 1/2. This is possible since B(s) is rapidly decreasing in vertical strips and

is also regular on the line <s = 1/2; the reason is that the term ζ(2s) from the

denominator has no zeros on this line. Therefore, the only poles of B(s) that we

encounter are

s = 1 (triple pole) and s = 1± 2iT (simple poles) (4.18)

By the theorem of residues, the right-hand side of 4.17 is

R.H.S. =16

|ξ(1 + 2iT )|2 ·1

2πi

∫

(1/2)

ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)

A1−sds

s− 1

+16

|ξ(1 + 2iT )|2 · [sum of residues] (4.19)

76

4.3.1 Computation of residues

The residues of B(s) at the simple poles s = 1± 2iT are

R1 := Ress=1 B(s) =(πA)−2iT

2√

πiTξ2(1 + 2iT ) · Γ(1

2+ 2iT )ζ(1 + 4iT )

Γ(1 + 2iT )ζ(2 + 4iT )(4.20)

and R2 := Ress=1−2iT B(s) = R1.

Therefore |R1 + R2| ≤ 2|R1|, and

|R1 + R2||ξ(1 + 2iT )|2 ≤

2|R1||ξ(1 + 2iT )|2 =

1

T√

π·∣∣∣∣Γ(1

2+ 2iT )ζ(1 + 4iT )

Γ(1 + 2iT )ζ(2 + 4iT )

∣∣∣∣

Using the estimate 3.16 for the Riemann zeta as well as Stirling’s asymptotic formula,

we obtain

R1 + R2

|ξ(1 + 2iT )|2 = O(T− 32 log T ), T →∞ (4.21)

The residue

R0 := Ress=1 B(s)

is slightly more complicated since s = 1 is a triple pole. Recall

B(s) =A1−s

s− 1

ξ2(s)ξ(s + 2iT )ξ(s− 2iT )

ξ(2s)(4.22)

The power series expansion of A1−sξ(s + 2iT )ξ(s− 2iT ) near s = 1 is

|ξ(1 + 2iT )|2

1 +(2<ξ′

ξ(1 + 2iT )− log A

) · (s− 1) +

+[2<ξ′′

ξ(1 + 2iT ) + 2

∣∣ξ′ξ

(1 + 2iT )∣∣2 − 4(log A)<ξ′

ξ(1 + 2iT ) + log2 A

] · (s− 1)2

+O((s− 1)3) (4.23)

77

Assume bi are the constants that occur in the power series expansion of ξ2(s)(s−1)ξ(2s)

near

s = 1:

ξ2(s)

(s− 1)ξ(2s)=

b−3

(s− 1)3+

b−2

(s− 1)2+

b−1

s− 1+ b0 + · · · (4.24)

We can determine the residue of B(s) at s = 1 by multiplying the previous two power

series, and after scaling we obtain

R0

|ξ(1 + 2iT )|2 = 2b−3<ξ′′

ξ(1 + 2iT ) + 2b−3

∣∣ξ′ξ

(1 + 2iT )∣∣2

+ (2b−2 − 4b−3 log A)<ξ′

ξ(1 + 2iT )

+ b−3 log2 A− b−2 log A + b−1 (4.25)

In order to estimate this quantity, we first write

ξ′

ξ(s) = −1

2log π +

1

2

Γ′

Γ(s

2) +

ζ ′

ζ(s)

and

ξ′′

ξ(s) =

1

4log2 π +

1

4

Γ′′

Γ(s

2) +

ζ ′′

ζ(s)

− 1

2(log π)

Γ′

Γ(s

2)− (log π)

ζ ′

ζ(s) +

Γ′

Γ(s

2)ζ ′

ζ(s)

Well-known estimates for the logarithmic derivatives of zeta and gamma functions

[Ti, 3.11.7]

Γ′

Γ(1

2+ iT ) = O(log T ),

ζ ′

ζ(1 + 2iT ) = O(log T ) (4.26)

and the second derivative

Γ′′

Γ(1

2+ iT ) = O(log2 T ),

ζ ′′

ζ(1 + 2iT ) = O(log2 T ) (4.27)

78

imply the same estimates for the logarithmic derivatives of ξ. We obtain

R0

|ξ(1 + 2iT )|2 = O(log2(AT )

)(4.28)

Eq. 4.21 and 4.28 allow us to complete the estimate for one of the terms of 4.19:

16

|ξ(1 + 2iT )|2 · [sum of residues] = O(log2(AT )

)(4.29)

and the implied constant is absolute.

Remark 4.1. Since we did not find one in the literature, we will give a proof of the

estimate 4.27 concerning the second derivative of the Riemann zeta at the end of this

chapter.

4.3.2 Evaluation of the shifted integral

To complete the proof of Theorem 4.13 we still have to evaluate the integral on the

right-hand side of 4.19. For s = 12

+ it,

B(s)

|ξ(1 + 2iT )|2 =(Aπ)

12−it

−12

+ it· Γ2(1

4+ it/2)

∏± Γ(1

4+ i(t/2± T ))

Γ(12

+ it)|Γ(12

+ iT )|2

× ζ2(12

+ it)∏± ζ(1

2+ i(t± 2T ))

ζ(1 + 2it)|ζ(1 + 2iT )|2

By Stirling’s formula, this quantity is bounded in absolute value by

A1/2

|ζ(1 + 2iT )|2exp π

2

(2T − | t

2− T | − | t

2+ T |)

(1 + |t|) 32 (1 + | t

2± T |) 1

4

∣∣∣∣ζ2(1

2+ it)

∏± ζ(1

2+ i(t± 2T ))

ζ(1 + 2it)

∣∣∣∣

Estimating the ζ-factors in the numerator with the subconvexity estimate (3.18), we

find an upper bound for the integrand

A1/2

|ζ(1 + 2it)||ζ(1 + 2iT )|2exp Ω(t, T )

(1 + |t|) ∏±(1 + |t± 2T |)1/4−θ

79

where Ω(t, T ) = π2(2T − | t

2− T | − | t

2+ T |). However, the factors ζ(1 + 2iT ) and

ζ(1 + it) from the denominators are harmless, since we have 1ζ(1+it)

= O(log t) (3.16).

The shifted integral is therefore bounded, in absolute value, up to an O(T ε) factor,

by

A1/2T−1/4+θ

∫ ∞

0

exp Ω(t, T )dt

(1 + |t|)(1 + |t− 2T |)1/4−θ(4.30)

We have already estimated integrals of this type in the previous chapter. We first

split the integral as∫

t>2Tdt +

∫t<2T

dt. In the range |t| > 2T the exponential decay

of the factor exp Ω(t, T ) takes over and we can easily see that integrating over this

range produces O(exp(−π2T )).

In the range |t| ≤ 2T we can ignore the exponential factor since, by the triangle

inequality, Ω(t, T ) ≤ 0. Therefore the main contribution is

A1/2T−1/4+θ

∫ 2T

0

(1 + |t|)−1(1 + |t− 2T |)− 14+θdt

¿ A1/2T−1/2+2θ+ε

Since the subconvexity exponent satisfies θ < 16, we find that the shifted integral is

O(A1/2T−1/6).

We recall the results of this section: eq. 4.17 and 4.19 state that

‖H‖22 ≤

16

|ξ(1 + 2iT )|2 ·1

2πi

∫

(2)

B(s)ds =16(R0 + R1 + R2)

|ξ(1 + 2iT )|2 + [shifted integral]

while 4.21 and 4.28 give bounds for the contribution of the residues:

R0 + R1 + R2

|ξ(1 + 2iT )|2 = O(log2(AT ))

At last, we saw that the shifted integral is O(A1/2T−1/6). Hence the estimate for the

L2 norm of H

‖H‖22 = O(log2(AT )) + O(A1/2T−1/6) (4.31)

80

with the implied constants absolute. Therefore, when A is fixed,

‖H‖2 = O(log T ), T →∞ (4.32)

and this finishes the proof of Proposition 4.3.

4.4 An Estimate for ζ ′′ζ (1 + it)

In this section we prove

Proposition 4.4. ζ′′ζ

(1 + it) = O(log2 t), when |t| → ∞.

The starting point is the Hadamard factorization formula for the completed zeta-

function

Λ(s) =s(s− 1)

2π−

s2 Γ(

s

2)ζ(s) (4.33)

which is entire and has order 1 :

Λ(s) =1

2eb0s

∏ρ

(1− s

ρ)es/ρ (4.34)

where the product is taken over the complex roots of the Riemann zeta-function

ρ = σ + iγ in the strip 0 < σ < 1 [Ti, 2.12]. Moreover, Λ(s) having order 1 implies

that the series∑

ρ

1

|ρ|1+ε

converges for any positive ε.

Taking the logarithmic derivative in 4.34 we obtain (b = b0 + 12log π)

ζ ′

ζ(s) =

d

dslog ζ(s) = b− 1

s− 1− 1

s− 1

2

d

dslog Γ(

s

2) +

∑ρ

1

s− ρ+

1

ρ

(4.35)

81

Differentiating once again, we obtain

d2

ds2log ζ(s) =

1

(s− 1)2+

1

s2− 1

4

d2

ds2log Γ(

s

2)−

∑ρ

1

(s− ρ)2(4.36)

From the Weierstrass factorization of the Gamma function

Γ(s) =e−cs

s

∞∏n=1

(1 +

s

n

)−1

es/n

it immediately follows that

d

dslog Γ(σ + it) = O(log t),

d2

ds2log Γ(σ + it) = O(log2 t) (4.37)

uniformly in σ. We also have 1s2 + 1

(s−1)2= O(1) trivially. To estimate the remaining

sum∑

ρ1

(s−ρ)2on the right-hand side of 4.36, we first need to develop an inequality

which is central in the method of Hadamard and de la Vallee Poussin. The next

lemma corresponds to equation [Ti, 3.8.5].

Lemma 4.5. For σ ≥ 1,

−<ζ ′

ζ(s) < A log t−

∑ρ

σ − β

|s− ρ|2

Proof. Identifying the real parts on both sides of equation 4.35 we obtain

−<ζ ′

ζ(s) = −b + <1

s+

1

s− 1+

1

2<Γ′

Γ(s

2)−

∑ρ

σ − β

|s− ρ|2 +β

|ρ|2

Since 0 < β < 1 ≤ σ, the terms of∑

ρ are positive. The lemma now follows from the

fact that Γ′Γ

( s2) = O(log t).

Using now the well-known estimate for the logarithmic derivative of ζ(s) [Ti, eq.

82

3.11.7]

ζ ′

ζ(σ + it) = O(log t), (4.38)

we obtain, as an immediate consequence, that

∑

ρ=β+iγ

σ − β

|s− ρ|2 = O(log t) (4.39)

uniformly in 1 ≤ σ ≤ 2, t > t0. We now write

∣∣∣∣∣∑

ρ

1

(s− ρ)2

∣∣∣∣∣ ≤∑

ρ

1

|s− ρ|2 =∑

|γ−t|<1

1

|s− ρ|2 +∑

|γ−t|≥1

1

|s− ρ|2

=: Σ1 + Σ2 (4.40)

a) In the range |γ − t| < 1 we appeal to the zero-free region of the Riemann zeta

1− C

log t≤ σ ≤ 1 (4.41)

from [Ti, Thm. 3.8], to ensure that

(1− β) log t ≥ C

Therefore

Σ1 ¿ log t∑

|γ−t|<1

1− β

|s− ρ|2

In view of 4.39, this is O(log2 t).

b) In the range |γ − t| ≥ 1 we have

Σ2 ≤∑

|γ−t|≥1

1

|γ − t|2 ≤ 2∑

|γ−t|≥1

1

1 + |γ − t|2

83

On the other hand, inequality 4.39 at s = 2 + it yields

∑

|γ−t|≥1

1

1 + |γ − t|2 ≤ 4∑

|γ−t|≥1

2− β

(2− β)2 + (γ − t)2¿ log t (4.42)

which implies Σ2 = O(log t).

This, together with the estimate for Σ1, shows that

d2

ds2log ζ(1 + it) =

ζ ′′

ζ(1 + it)− (

ζ ′

ζ)2(1 + it) = O(log2 t)

Since we have ζ′ζ(1 + it) = O(log t) already, we arrive at the desired result.

84

Chapter 5

A Family Sum

5.1 Preliminaries

In this chapter we study the arithmetic quantity

Disc.(∞; T ) :=∑

φ

∣∣∣⟨E2(

1

2+ iT ), φ

⟩∣∣∣2

(5.1)

This quantity is converted, via the Luo-Sarnak triple product formula of Section

4.2.1, into an average of Maass-Hecke L-functions, hence the term ’family sum’. By

rescaling T 7→ T/2 for convenience,

Disc.(∞; T/2) =∑

φ

|w(tφ, T )|2L2(1

2, φ)

∣∣L(1

2+ iT )

∣∣2 (5.2)

where φ varies over the set of L2-normalized Hecke-Maass forms and the weight

w(tφ, T ) is defined in Theorem 4.1.

Averages (or moments) of automorphic L-functions were traditionally used for

proving subconvex estimates for an individual object. The key step in this approach

is finding the appropriate family for the specific L-function, and then using trace

formulae to evaluate the family average. By positivity, this would lead to an estimate

85

for the original L-function.

In our case, the family arises naturally from the spectral expansion of the L4

norm ‖EA(12

+ iT )‖4. The novelty of the present situation is that the estimate that

we obtain by using the family method (Corollary 5.2) is sharp (see the explanation

from Section 1.2). This is in contrast, for example, with the case of subconvexity,

where the estimates obtained by the family method are never sharp.

5.1.1 An asymptotic formula for the weight

We return to the quantity of 5.2; the sum is weighted by

w(tφ, T ) := |w(tφ, T )|2 (5.3)

and w was defined in Theorem 4.1. We have the explicit formula

w(tφ, T ) =|ρφ(1)|2

4|ζ(1 + iT )|4|Γ(1

4+ i

tφ2)|4 ∏

± |Γ(14

+ itφ±T

2)|2

|Γ(12

+ iT2)|4

=παφ

4|ζ(1 + iT )|4|Γ(1

4+ i

tφ2)|4 ∏

± |Γ(14

+ itφ±T

2)|2

|Γ(12

+ itφ)|2|Γ(12

+ iT2)|4 (5.4)

with the normalizing factor αφ =|ρφ(1)|2cosh πtφ

. Recall property (2.6):

t−εφ ¿ αφ ¿ tεφ, ∀ε > 0 (5.5)

By Stirling’s formula (3.14),

w(tφ, T ) =2π2αφ

|ζ(1 + iT )|4exp Ω(tφ, T )

(1 + tφ)∏±(1 + |tφ ± T |)1/2

1 + O

( 1

1 + |tφ − T |)

(5.6)

with the exponent Ω given by

Ω(tφ, T ) =π

2(2T − |tφ + T | − |tφ − T |) (5.7)

86

We thus obtain a preliminary estimate for the weight

w(tφ, T ) ³ αφ

|ζ(1 + iT )|4 ×

t−1φ T− 1

2 (1 + |T − tφ|)− 12 , tφ < T

t− 3

2φ (1 + |tφ − T |)− 1

2 exp(−π|tφ − T |), tφ ≥ T

We remark that Ω(tφ, T ) ≤ 0 by the triangle inequality. Also, exp Ω(tφ, T ) is rapidly

decreasing when tφ ≥ T + log2 T .

5.1.2 The Main Theorem

The main result of this chapter is a sharp bound on Disc.(∞; T/2). The asymptotic

analysis is left for the Appendix.

Theorem 5.1. Let ε > 0, then

Disc.(∞; T/2) = O(T ε), T →∞ (5.8)

As an immediate corollary, we obtain part (a) of Theorem L4 :

Corollary 5.2. ‖E(12

+ iT )‖4 = O(T ε), ∀ε > 0

5.1.3 Decomposition after suitable ranges

We split the sum expressing Disc.(∞; T/2) in 5.2 into three parts as follows

∑

φ

=∑

0≤tφ<P1

+∑

P1≤tφ≤P2

+∑

P2≤tφ

=: Σbulk + Σtrans + Σ3 (5.9)

with the cut-off points

P1 = T − 1

4T 1−4δ, P2 = T +

1

4T 1−4δ (5.10)

87

and δ a small positive number, to be specified later. (Essentially, ’δ = ε’.)

a) As we already mentioned before, the exponential decay of the weight kicks in

precisely in the range of Σ3. Therefore, the contribution of this sum to Disc.(∞; T/2)

is O(T−N), ∀N ≥ 1, i.e. negligible.

b) The range 0 ≤ tφ < P1 of Σ1 will be referred to as the bulk range, and it

turns out that Σ1 has the main contribution to the total sum Σφ. In this range, the

weight is roughly of size T−2, therefore we need to prove the estimate

∑tφ³T

L2(1

2, φ)

∣∣L(1

2+ iT )

∣∣2 ¿ T 2+ε (5.11)

which is similar to the spectral fourth moment of the Maass-Hecke L-functions.

The estimate 5.11 can be interpreted as an instance of the Generalized Lindelof

Hypothesis (GLH) on average. GLH predicts that each term in the sum 5.11 is O(T ε)

while, by Weyl’s law, there are roughly T 2 terms in the sum. We remark that subcon-

vex bounds for the individual summands would fail to produce the desired estimate.

We thus need to exploit the fact that we average over φ, and we shall do that through

the large sieve inequality of Deshouillers and Iwaniec. In order to apply the large

sieve, we first have to replace the L-functions in questions by Dirichlet polynomials of

finite length. This is the principle of the approximate functional equation. Moreover,

when φ is in the bulk range, L(12, φ) and L(1

2+ iT ) have essentially the same length,

which makes the large sieve applicable.

c) The transition range P1 ≤ tφ ≤ P2 of Σ2 has length 12T 1−4δ, and the large

sieve inequality does not yield good results on a short interval. We first divide the

transition range into dyadic ranges |tφ − T | ∼ H, where the weight is roughly of size

T− 32 H− 1

2 . Therefore, we essentially need to prove the estimate

∑T−H≤tφ≤T+H

L2(1

2, φ)

∣∣L(1

2+ iT )

∣∣2¿ T 3/2+εH1/2 (5.12)

88

uniformly in 1 ≤ H ¿ T 1−4δ. Since there are roughly TH terms in this sum, the

estimate corresponding to GLH on average is O(T 1+εH). This is a harder problem

than 5.11, since the average is over a short interval. However, in this situation we

can settle for less since, as remarked, O(T32+εH

12 ) is acceptable.

The proof of 5.12 proceeds with lengthening the summation (exploiting positivity),

and then with an application of Kuznetsov’s trace formula. The outcome is a non-

diagonal term of lower order, at the expense of a slightly larger, yet admissible,

diagonal term. This phenomenon resembles the proof of the fourth moment estimate

for the Riemann zeta-function, discussed in Section 3.4.2.

5.2 Main Ingredients

5.2.1 Approximate functional equation

Originally developed as a tool to evaluate the Riemann zeta-function ζ(s) in the criti-

cal strip, the method of the approximate functional equation (AFE) was subsequently

generalized to the larger class of automorphic L-functions, with important applica-

tions in the theory of moments and subconvex estimates. The term was coined by

Hardy and Littlewood in their 1921 paper [H-L1].

The version of AFE that we use is due to G. Harcos and it works for a general

automorphic form on GL(n). However, we will only state it for n = 2. For reference,

see [Ha, Thm. 2.5].

Proposition 5.3 (Harcos). Suppose π is an automorphic form on GL(2) (full level)

with unitary central character. Suppose that the L-factor at the infinite place is given

by

L(s, π∞) = ΓR(s− µ1)ΓR(s− µ2)

89

Let C = C(π) be the analytic conductor of π:

C =1

4π2

∣∣∣12− µ1

∣∣∣∣∣∣12− µ2

∣∣∣

Let η = minj=1,2

∣∣12− µj

∣∣. Then, for a smooth function f1 on (0,∞), rapidly decreasing

at ∞, which satisfies

f1(x) + f1(1/x) = 1

we have:

L(1

2, π

)=

∞∑n=1

λπ(n)√n

f1

(nX√C

)+ γπ

∞∑n=1

λπ(n)√n

f1

( n

X√

C

)

+ Oε,f1(η−1C

14+ε), ∀ε > 0 (5.13)

uniformly in 1/A ≤ X ≤ A, A an absolute constant. Here

γπ =L(1

2, π∞)

L(12, π∞)

is a complex number that depends on π only, |γπ| = 1. The implied constant in the

error term depends on ε and f1.

We made a minor modification by introducing the extra parameter X, but this

does not change the argument in [Ha]. From now on we will denote by X ³ Y a

relation of the type 1/A ≤ X/Y ≤ A, with an absolute constant A > 0. We saved

the letter f for the function f(x) = f1(2πx) for convenience.

In our application of this theorem we will always take a smooth function f1 of

compact support in [0,∞), hence the Dirichlet series approximating L(12, φ) will in

fact be a Dirichlet polynomial with ¿ √C terms. This means that the length of the

approximate functional equation is ”square root of the conductor”.

90

Convexity estimate. As an immediate corollary of the approximate functional

equation, we have the convexity estimate

L(1

2, π) = O

(C(π)

14+ε

), ∀ε > 0 (5.14)

5.2.2 Kuznetsov’s trace formula

Let h(r) be an even analytic function in a horizontal strip |=r| ≤ 12

+ δ, satisfying

the growth condition

|h(r)| ¿ (1 + |r|)−2−δ

where δ is an arbitrarily small positive constant. Then the following identity holds

for any m, n ≥ 1

∑

φ

αφh(tφ)λφ(m)λφ(n) +1

π

∫ ∞

−∞

τir(m)τir(n)

|ζ(1 + 2ir)|2h(r)dr

=δm,n

π2

∫ ∞

−∞t tanh(πt)h(t)dr +

∞∑c=1

S(m,n; c)

ch+

(4π√

mn

c

)(5.15)

Here τir(m) are the Fourier coefficients of the Eisenstein series E(z, 12

+ ir), and

h+(X) :=2i

π

∫ ∞

−∞J2ir(X)h(r)

rdr

cosh πr

is the Bessel transform. For reference, see [Iw, Thm. 9.3], [D-I, Prop. 2].

5.2.3 Spectral large sieve

As an application of the trace formula, Deshouillers and Iwaniec developed a spectral

large sieve inequality, which is an average estimate for a general bilinear form in the

Hecke eigenvalues. Theorem 2 of [D-I] states the following: assume T ≥ 1, N ≥ 1/2

and ε > 0 are positive real numbers, and a(n) are arbitrary complex numbers, for

91

N < n ≤ 2N . We have

∑0≤tφ≤T

αφ

∣∣∣∣∣∑

N≤n<2N

a(n)λφ(n)

∣∣∣∣∣

2

¿ (T 2 + N1+ε)∑

N≤n<2N

|a(n)|2 (5.16)

and the implied constant depends on ε only.

We remark that, in view of the Hoffstein-Lockhart and Iwaniec inequalities (2.6)

satisfied by the normalizing factor αφ, the estimate 5.16 holds, up to an extra-factor

of T ε, with or without αφ on the left-hand side.

5.2.4 Spectral second moment

We find it useful to appeal later on to the following result of Kuznetsov and Motohashi,

regarding the mean square of central values of L-functions. When φ ranges over the

Hecke-Maass forms of SL(2,Z)\H, the following asymptotic formula holds

∑tφ≤T

αφL2(

1

2, φ) =

2T 2

π2

(log

T

2+ γ − 1

2

)+ O(T log6 T ) (5.17)

For reference, see [Mot].

5.3 The Bulk Range

In this section we analyze the weighted average

Σbulk =∑

1≤tφ≤P1

w(tφ, T )L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (5.18)

where P1 = T − 14T 1−4δ and 0 < δ < 1 is fixed until the end of the chapter.

92

Dyadic subdivision. Let H be a parameter such that 1 ≤ H ≤ 12P1. We consider

the dyadic interval H ≤ tφ < 2H. By 5.6, when φ is in this range, the weight satisfies

w(tφ, T ) ³ αφ

|ζ(1 + iT )|4 · T−1/2H−1(1 + |T − tφ|)−1/2

¿ αφ

|ζ(1 + iT )|4 · T−1+2δH−1

Therefore, summing over all the dyadic subintervals of the bulk range, we have

Σbulk ¿ 1

|ζ(1 + iT )|4 ·∑H

T−1+2δH−1SH (5.19)

where

SH :=∑

H≤tφ<2H

αφL2(

1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (5.20)

The main result of this section is the following

Proposition 5.4. The following estimate holds uniformly in 1 ≤ H ≤ 12P1

∑H≤tφ<2H

L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 ¿ T 1+εH (5.21)

for any positive ε. The implied constant depends on ε only.

We remark that the statement of the proposition is equivalent to

SH = O(T 1+εH) (5.22)

in view of the estimate 5.5 satisfied by the normalizing factor αφ.

Returning now to inequality 5.19, if we take into account that |ζ(1 + iT )|−1 =

O(T ε),∀ε > 0 we obtain, as an immediate consequence of the proposition, the estimate

Σbulk = O(T 3δ) (5.23)

93

which is acceptable for our purposes. (Note that the number of dyadic intervals

[H, 2H) in the bulk range is less than log T .)

We begin the proof of Proposition 5.4 with the Cauchy-Schwartz inequality

∑tφ∼H

L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 ≤[ ∑

tφ∼H

L4(1

2, φ)

]1/2[ ∑tφ∼H

∣∣L(1

2+ iT, φ)

∣∣4]1/2

(5.24)

The next step is to obtain upper bounds for each of the sums on the right, separately.

We proceed first by reducing the problem, via the approximate functional equation,

to that of an average estimate of bilinear forms in the Hecke coefficients; then, we

apply the spectral large sieve.

5.3.1 Part 1:∑

H≤tφ≤2H L4(12 , φ)

Approximate formulae

Following Theorem 5.3, we find that the conductor of L(12, φ) is

1

4π2

∣∣∣12

+ itφ

∣∣∣2

=1

4π2λφ, (5.25)

with λφ the Laplace eigenvalue of φ. In the dyadic range H ≤ tφ ≤ 2H, we clearly

have

H2 ≤ λφ ≤ 3H2 (5.26)

The corresponding η of Theorem 5.3 satisfies

η =∣∣∣12

+ itφ

∣∣∣ ≥ H (5.27)

Therefore, by the approximate functional equation, we have in this case

L(1

2, φ) =

∞∑n=1

λφ(n)√n

f( nX√

λφ

)+ γφ

∞∑n=1

λφ(n)√n

f( n

X√

λφ

)+ Oε(H

−1/2+ε) (5.28)

94

with |γφ| = 1, X ³ 1, and f a smooth function of compact support in [0,∞) satisfying

f(t)+f(1/4π2t) = 1. This makes each of the sums on the right-hand side finite, with

¿ H number of terms. Moreover, 5.28 is uniform in X ³ 1.

Leveling the argument

In order to apply the spectral large sieve, we need to make sure that the Dirichlet

polynomial in the AFE depends on φ through the Hecke eigenvalues λφ(n) only. We

manage to do so by averaging over the parameter X, which was introduced solely

for this purpose. Since eq. 5.28 is uniform in X, we can integrate with respect to

1 ≤ X ≤ e to obtain, after a change of variable,

∣∣L(1

2, φ)

∣∣ ≤∫ e

1

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f( nX√

λφ

)∣∣∣∣∣dX

X+

∫ e

1

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f( n

X√

λφ

)∣∣∣∣∣dX

X

+ Oε(H−1/2+ε)

=

∫ eH

λ1/2φ

H

λ1/2φ

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f(nX

H

)∣∣∣∣∣dX

X+

∫ eλ1/2φH

λ1/2φH

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f( n

XH

)∣∣∣∣∣dX

X

+ Oε(H−1/2+ε)

By 5.26, 13≤ λ

1/2φ

H≤ 3. Since the integrand is positive, we can extend the range of

integration to obtain an upper bound:

∣∣L(1

2, φ)

∣∣ ≤ 2

∫ 3e

1/3e

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f(nX

H

)∣∣∣∣∣dX

X+ Oε(H

−1/2+ε) (5.29)

Using Holder’s inequality and summing over φ in the dyadic range, we are led to

∑H≤tφ≤2H

∣∣L(1

2, φ)

∣∣4 ¿∫ 3e

1/3e

∑H≤tφ≤2H

∣∣∣∣∣∞∑

n=1

λφ(n)√n

f(nX

H

)∣∣∣∣∣

4dx

X+ Oε(H

ε) (5.30)

95

The error term is explained by Weyl’s law, since the sum∑

tφ∼H contains O(H2)

terms. We anticipate that we shall obtain an upper bound for the main term which

will render the error term O(Hε) redundant. Hence we can ignore it from now on.

By the mean-value theorem applied to the integral on the right-hand side of 5.30,

there exists ξ ∈ (1/3e, 3e) such that

∑H≤tφ<2H

∣∣L(1

2, φ)

∣∣4 ¿∑

H≤tφ<2H

∣∣∣∣∣∑

n

λφ(n)√n

f(nξ

H

)∣∣∣∣∣

4

(5.31)

Employing the Hecke relations, we transform the right-hand side into a sum of bilinear

forms in Hecke coefficients.

Squaring. Hecke relations

The identities (2.4) satisfied by the Hecke eigenvalues λφ(n) show that squaring a

Dirichlet polynomial produces another Dirichlet polynomial of squared length

[∑n

λφ(n)√n

f(nξ

H

)]2

=∑

n

λφ(n)a(n)√n

(5.32)

with coefficients

a(n) =∑

d≥1

1

d

∑

n=kl

f(dkξ

H

)f(dlξ

H

)(5.33)

independent of φ. Since f has compact support at infinity, a(n) = 0 unless dkξH

, dlξH¿

H, and implicitly n ¿ H2. Moreover, since f is bounded, a(n) is bounded up to an

absolute constant by∑

n=kl

∑d¿H

1d. Trivially,

a(n) = O(τ(n) log H) (5.34)

uniformly in n, and a(n) = 0 unless n ¿ H2.

96

Applying the spectral large sieve

We are now in a position to apply the large sieve inequality of Deshouillers and

Iwaniec; the right-hand side of 5.31 is

∑H≤tφ<2H

∣∣∣∣∣∑

n¿H2

λφ(n)a(n)√n

∣∣∣∣∣

2

¿ H2+ε∑

n¿H2

|a(n)|2n

(5.35)

By 5.34 the right-hand side is bounded, up to a constant (depending on ε), by

H2+ε ·∑

n¿H2

τ 2(n)

n¿ H2+2ε

since τ(n) = O(nε). Combined with 5.31, this yields

∑H≤tφ<2H

L4(1

2, φ) = O(H2+ε) (5.36)

for any ε > 0.

5.3.2 Part 2:∑

H≤tφ≤2H

∣∣L(12 + iT, φ)

∣∣4

We treat this case in a completely analogous manner.

Approximate formulae

The conductor of L(12

+ iT, φ) is 14π2 Cφ, where

Cφ =∣∣∣12

+ i(T + tφ)∣∣∣∣∣∣12

+ i(T − tφ)∣∣∣ (5.37)

Since H ≤ tφ < 2H and H ≤ 12P1 = T − 1

4T 1−4δ, Cφ satisfies the inequalities

1

4T 2−4δ ≤ Cφ ≤ 2T 2 (5.38)

97

The corresponding η from the error term satisfies

η =∣∣∣12

+ i(T − tφ)∣∣∣ ≥ 1

4T 1−4δ (5.39)

The approximate functional equation obtained from Theorem 5.3 is:

L(1

2+ iT, φ) =

∞∑n=1

λφ(n)

n1/2+iTf( nX√

Cφ

)+ γφ(T )

∞∑n=1

λφ(n)

n1/2−iTf( n

X√

Cφ

)

+ Oε(T− 1

2+4δ+ε). (5.40)

for any positive ε. The error term is uniform in X ³ 1, |γφ| = 1, and f has compact

support at infinity.

Leveling the argument

We execute the same integral average over the parameter X as in the previous section.

Once again, the goal is to eliminate the φ-dependence in the argument of the test

function f . We have

∣∣L(1

2+ iT, φ)

∣∣ ≤∫ e

1

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2+iTf( nX√

Cφ

)∣∣∣∣∣dX

X+

∫ e

1

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2−iTf( n

X√

Cφ

)∣∣∣∣∣dX

X

+ Oε(T− 1

2+4δ+ε) (5.41)

By a change of variable, the right-hand side is

∫ eT

C1/2φ

T

C1/2φ

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2+iTf(nX

T

)∣∣∣∣∣dX

X+

∫ T

C1/2φ

T

eC1/2φ

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2−iTf(nX

T

)∣∣∣∣∣dX

X(5.42)

98

plus the same error term. By 5.38, 1√2≤ T

C1/2φ

≤ 2T 2δ. Since the integrand is positive

we obtain, by extending the range of integration,

∣∣L(1

2+ iT )

∣∣ ≤ 2

∫ bT 2δ

a

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2+iTf(nX

T

)∣∣∣∣∣dX

X+ Oε(T

− 12+4δ+ε) (5.43)

where a = 1e√

2and b = 2e for convenience. Since 5.43 is uniform in tφ ∼ H, we apply

Holder’s inequality and then sum over φ to obtain

∑H≤tφ<2H

∣∣L(1

2+ iT, φ)

∣∣4 ¿ (1 + log4(T 2δ)

)×

×∫ bT 2δ

a

∑H≤tφ<2H

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2+iTf(nX

T

)∣∣∣∣∣

4dX

X+ O(H2T−2+16δ+ε) (5.44)

The error term comes from multiplying the fourth power of the error term from 5.43

by H2, the number of terms in the sum∑

tφ∼H . Since H ≤ T , the error term is in

fact O(T 16δ+ε), which is redundant.

By the mean-value theorem applied to the integral on the right-hand side, there

exists ξ ∈ (a, bT 2δ) such that:

∑H≤tφ<2H

∣∣L(1

2+ iT, φ)

∣∣4 ¿ (1 + log5(T 2δ)

) ·∑

H≤tφ<2H

∣∣∣∣∣∞∑

n=1

λφ(n)

n1/2+iTf(nξ

T

)∣∣∣∣∣

4

(5.45)


As before, we begin by squaring the Dirichlet polynomial

∣∣∣∣∣∑

n

λφ(n)

n1/2+iTf(nξ

T

)∣∣∣∣∣

2

=∑

n

λφ(n)bT (n)√n

(5.46)

99

with the coefficients bT (n) given by

bT (n) =∑

d≥1

1

d

∑

n=kl

(k

l

)iTf(dkξ

T

)f(dlξ

T

)(5.47)

Once again, bT (n) = 0 unless n ¿ T 2. Moreover, for n in this range we can bound

bT (n) trivially by∑

n=kl

∑d¿T

1d, and obtain the estimate

bT (n) = O(τ(n) log T ) (5.48)

Applying the spectral large sieve

We reduced the problem of estimating an average of L-functions to that of estimating

an average of bilinear forms in Hecke coefficients: the right-hand side of 5.45 equals,

in view of 5.46,∑

H≤tφ<2H

∣∣∣∣∣∑

n¿T 2

λφ(n)bT (n)√n

∣∣∣∣∣

2

(5.49)

By the large sieve inequality (5.16) this quantity is bounded by

T ε(H2 + T 2) ·∑

n

|bT (n)|2n

for any ε > 0. Using the upper bound 5.48, the right-hand side is certainly less than

T 2+ε ·∑

n¿T 2

τ 2(n)

n¿ T 2+2ε

Combined with the inequality 5.45, this yields

∑H≤tφ<2H

∣∣L(1

2+ iT, φ)

∣∣4 = O(T 2+ε), ∀ε > 0 (5.50)

100

This estimate, together with 5.36 and the Cauchy-Schwartz inequality 5.24, leads to

the desired result of Proposition 5.4:

∑H≤tφ<2H

L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 = O(T 1+εH)

As remarked in the beginning of this section, this in turn implies 5.23, which is what

we set out to prove in this section.

5.3.3 Average over a finite interval

In summing over 1 ≤ tφ ≤ P1 in the definition of Σbulk, we tacitly assumed that the

first Laplace eigenvalue on Γ\H satisfies the lower bound

λ1 >1

4(5.51)

While this is known to be true for Γ = SL(2,Z), it is not true for an arbitrary

congruence subgroup. Therefore, we consider appropriate a separate treatment of the

sum

Σ0 =∑

0<λφ≤ 14

L2(1

2, φ)

∣∣L(1

2+ iT )

∣∣2 (5.52)

In this case, we can use Meurman’s [Me] subconvexity bound (in the t aspect) for a

GL(2) automorphic L-function:

L(1

2+ iT, φ) = O(T

13+ε), T →∞ (5.53)

Since there are only finitely many Hecke-Maass forms φ with 0 < λφ ≤ 14, the preced-

ing estimate is uniform in these φ’s. On the other hand, w(tφ, T ) = O(T−1+ε) in this

101

range, hence

Σ0 ¿ T− 13+ε ·

∑

0<λφ≤ 14

L2(1

2, φ) = O(T−1/3+ε)

Therefore this range does not affect the estimate obtained at 5.23.

5.4 The Transition Range

In this section we analyze the transition range P1 ≤ tφ ≤ P2. Its contribution to the

family sum Disc.(∞; T/2) is given in equation 5.9:

Σtrans =∑

P1≤tφ<P2

w(tφ, T )L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2

Recall that

P1 = T − 1

4T 1−4δ, P2 = T +

1

4T 1−4δ (5.54)

and δ > 0 remains fixed until the end of this chapter. The aim of the current section

is to prove the following estimate:

Σtrans = O(T 3δ) (5.55)

First, we eliminate the range T −1 ≤ tφ ≤ T +1 by appealing to convexity estimates.


+ iT, φ) is O(T ) in this range, hence the convexity estimate

(5.14) gives

L(1

2+ iT, φ) = O(T 1/4+ε) (5.56)

uniformly in T − 1 ≤ tφ ≤ T + 1. Moreover, in this range, the weight satisfies

w(tφ, T ) = O(T− 32+ε). Summing over T − 1 ≤ tφ ≤ T + 1, we have

∑T−1≤tφ≤T+1

w(tφ, T )L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 ¿ T−1+ε ·∑

T−1≤tφ≤T+1

L2(1

2, φ)

102

In view of the spectral second moment 5.17, the right-hand side is O(T ε), for any

ε > 0. Therefore the range T − 1 ≤ tφ ≤ T + 1 does not affect estimate 5.55.

Dyadic subdivision. We further divide the transition range 1 ≤ |tφ−T | ≤ 14T 1−4δ

into dyadic intervals of the form

H ≤ |tφ − T | < 2H, (5.57)

with 1 ≤ H ≤ 18T 1−4δ. The reason is that on such intervals the weight satisfies the

uniform estimate

w(tφ, T ) ³ αφ

|ζ(1 + iT )|4 · T− 3

2 H− 12 = O(T− 3

2+εH− 1

2 ) (5.58)

and hence Σtrans is bounded, up to an absolute constant, by

1

|ζ(1 + iT )|4 ·∑H

T− 32 H− 1

2 ·∑

H≤|tφ−T |≤2H

αφL2(

1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (5.59)

It is convenient to keep track of the factor αφ, as it occurs naturally in the trace

formula of Kuznetsov. Since |ζ(1 + iT )|−1 = O(T ε) for any ε > 0, and the number of

dyadic intervals is O(log T ), it means that in order to prove 5.55 it is enough to show

that each dyadic sum

S(2)H :=

∑

H≤|tφ−T |≤2H

αφL2(

1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (5.60)

is uniformly O(T32+2δH

12 ). In this way, we reduce 5.55 to proving the following

Proposition 5.5. Assume δ > 0 and H is such that 1 ≤ H ≤ 18T 1−4δ. Then

S(2)H = O(T

32+2δH

12 ) (5.61)

103

The rest of the section is concerned with proving this proposition.

5.4.1 Approximate formulae

Once again, the starting point is reducing the L-functions in question to Dirichlet

polynomials of finite length.

Recall that φ is a Hecke-Maass form with Laplace eigenvalue λφ = 14

+ t2φ and

H ≤ |tφ − T | ≤ 2H, with 1 ≤ H ≤ 18T 1−4δ. Hence

1

4T 2 ≤ λφ ≤ T 2 (5.62)

The conductor of L(12, φ) is simply 1

4π2 λφ, and the corresponding η of Theorem

5.3 is η =√

λφ ≥ T/2. The approximate functional equation gives

L(1

2, φ) =

∞∑n=1

λφ(n)√n

f( nX√

λφ

)+ γφ

∞∑n=1

λφ(n)√n

f( n

X√

λφ

)+ Oε(T

− 12+ε) (5.63)

uniformly in X ³ 1. Here |γφ| = 1 and, as before, f is a smooth function of compact

support at infinity satisfying f(x) + f(1/4π2x) = 1.


+ iT, φ) is 14π2 Cφ, with Cφ =

∣∣12

+ i(tφ − T )∣∣∣∣1

2+ i(tφ + T )

∣∣,while η =

∣∣12

+ i(tφ − T )∣∣ ≥ H. The fact that H ≥ 1 ensures that

TH ≤ Cφ ≤ 5TH (5.64)

and hence Cφ/TH ³ 1. The approximate functional equation in this case gives

L(1

2+ iT, φ) =

∞∑n=1

λφ(n)

n1/2+iTf( nY√

Cφ

)+ γφ(T )

∞∑n=1

λφ(n)

n1/2−iTf( n

Y√

Cφ

)

+ Oε(T14+εH− 3

4 ) (5.65)

uniformly in Y ³ 1, with |γφ(T )| = 1.

104

Remark 5.1. Since f has compact support at infinity, the two series of 5.65 have

finite length, as the summand vanishes for n À √TH. Therefore L(1

2+ iT, φ) has

length√

TH when φ is in the dyadic range corresponding to H. By squaring, we will

see that∣∣L(1

2+ iT, φ)

∣∣2 has length TH. Similarly, we say that L2(12, φ) has length T 2.

For reasons of clarity, we use the notation

K :=√

TH (5.66)

Dirichlet polynomials. Let us denote by LX(12, φ) the Dirichlet polynomial of

5.63, so that

L(1

2, φ) = LX(

1

2, φ) + O(T− 1

2+ε)

and hence

L2(1

2, φ) ≤ 2

∣∣LX(1

2, φ)

∣∣2 + O(T−1+ε) (5.67)

uniformly in X ³ 1.

Similarly, we have

∣∣L(1

2+ iT, φ)

∣∣2 ≤ 2∣∣LY (

1

2+ iT, φ)

∣∣2 + O(T12+εH− 3

2 ) (5.68)

where LY (12

+ iT, φ) is the Dirichlet polynomial from 5.65, and Y ³ 1.

Multiplying the previous two equations and summing over φ in the dyadic range,

we obtain

S(2)H =

∑

|T−tφ|∼H

αφL2(

1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 ≤ 4∑

φ

αφ

∣∣LX(1

2, φ)

∣∣2∣∣LY (1

2+ iT, φ)

∣∣2

+ Error (5.69)

105

with the error term given, up to a constant, by

T12+εH− 3

2 ·∑

φ

L2(1

2, φ) + T−1+ε ·

∑

φ

∣∣L(1

2+ iT, φ)

∣∣2 + T− 12+εH− 3

2 ·∑

φ

1 (5.70)

and the sum is over H ≤ |tφ − T | ≤ 2H.

5.4.2 Error term estimate

We use different methods to evaluate the sums of 5.70.

a) Weyl’s law implies that each of these sums has O(T 1+εH) terms.

b) We use the spectral second moment (5.17) to evaluate the first sum. Since

T > H ≥ 1, we have

∑

H≤|tφ−T |≤2H

L2(1

2, φ) = O(T 1+εH), ∀ε > 0

c) When H ≤ |tφ − T | ≤ 2H, the convexity estimate for L(12

+ iT, φ) is

L(1

2+ iT, φ) = O

((TH)

14+ε

)

and hence

∑

H≤|tφ−T |≤2H

∣∣L(1

2+ iT, φ)

∣∣2 ¿ (TH)12+ε ·

∑

H≤|tφ−T |≤2H

1 = O((TH)

32+ε

)

Combining these estimates with 5.70, we obtain an upper bound for the error term:

Error = O(T32+εH− 1

2 ) + O(T12+εH

32 ) + O(T

12+εH− 1

2 ) = O(T32+εH

12 ) (5.71)

106

for any ε > 0. In view of our goal (Proposition 5.5), we can ignore this error term

from now on.

5.4.3 Leveling the argument

We return to eq. 5.69 which gives uniform estimates in X,Y ³ 1; we integrate over

X and Y in order to level the argument of the Dirichlet polynomials LX(12, φ) and

LY (12

+ iT, φ). By Cauchy-Schwartz inequality S(2)H is bounded, up to a constant, by

∑

H≤|tφ−T |<2H

αφ

∫ e

1

∫ e

1

∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX√

λφ

)∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf( kY√

Cφ

)∣∣∣∣∣

2dX

X

dY

Y(5.72)

After a change of variable, this is

∑

H≤|tφ−T |<2H

αφ

∫ eT

λ1/2φ

T

λ1/2φ

∫ eK

C1/2φ

K

C1/2φ

∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX

T

)∣∣∣∣∣

2

×

×∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf(kY

K

)∣∣∣∣∣

2dX

X

dY

Y

with K =√

TH (5.66). By 5.62 and 5.64, 1 ≤ T

λ1/2φ

≤ 2 and 1√5≤ K

C1/2φ

≤ 1. Therefore

we can extend the limits of integration (the integrand is positive) to obtain the upper

bound

∫ 2e

1

∫ e

1√5

∑

φ

αφ

∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX

T

)∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf(kY

K

)∣∣∣∣∣

2dX

X

dY

Y(5.73)

By the mean-value theorem applied to the double integral, there exist X ∈ (1, 2e)

and Y ∈ ( 1√5, e), independent of φ, such that

S(2)H ¿ J (5.74)

107

with

J =∑

H≤|tφ−T |≤2H

αφ ·∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX

T

)∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf(kY

K

)∣∣∣∣∣

2

(5.75)

5.4.4 Lengthening the summation

The purpose of this section is to prove the estimate

J = O(T32+2δH

12 ) (5.76)

This will finish the proof of Proposition 5.5.

Remark 5.2. In this section the usefulness of the cut-off points P1,2 = T ± 14T 1−4δ

will become apparent: it allows enough room for over-summing, and is relevant in a

lemma on the Bessel transform.

We define

∆ = T12+δH

12 (5.77)

This parameter has the following properties:

i) H ≤ ∆

ii) ∆ ≤ 1

2T 1−δ

iii) T32 H

12 = T 1−δ∆ (5.78)

We emphasize these properties as they are relevant, the first in lengthening the sum-

mation, and the last two as technical conditions in a lemma on the integral Bessel

transform.

The next step is to extend the summation range in J (eq. 5.75) from |tφ−T | ∼ H

108

to |tφ − T | ¿ ∆, by introducing a smooth weight. Let

h∆(t) =1

2 tanh(πt)

t

T·(

exp(−π

(t− T

∆

)2)+ exp

(−π(t + T

∆

)2))(5.79)

This function satisfies the conditions required by Kuznetsov’s trace formula: it is

even, regular in the horizontal line |=t| < π, and rapidly decreasing in <t.

Moreover, h∆ is positive on the real line and is bounded from below by an absolute

constant on an interval larger than the dyadic range [T − 2H, T −H]:

h∆(t) ≥ 2−√2

4e−4π when |t− T | ≤ 2∆ (5.80)

We can now extend th range in the sum on the right-hand side of 5.75, to obtain an

upper bound for J

J ≤ 4e4π

2−√2·∑

all φ

αφh∆(tφ)

∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX

T

)∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf(kY

K

)∣∣∣∣∣

2

(5.81)

where now φ ranges over all the Hecke-Maass forms spanning the discrete spectrum

L2d(Γ\H). We denote by J∞ the sum on the right-hand side.

Remark 5.3. This step is analogous to the lengthening method used in estimating

the fourth moment of Riemann zeta, in section 3.4.2. We remark that the terms

corresponding to |tφ − T | À H no longer represent L-functions, yet they are positive

and contribute to the upper bound. We can view this as an embedding of the sum

of L-functions into a ’fake’ family. The right-hand side of 5.81 has the advantage of

being a complete sum (we sum over all φ), hence we can apply the trace formula.

109


We first square the Dirichlet polynomials

∣∣∣∣∣∞∑

m=1

λφ(m)

m1/2f(mX

T

)∣∣∣∣∣

2

=∑m,n

λφ(m)λφ(n)√mn

f(mX

T

)f(nX

T

)

=∑m

λφ(m)a(m)√m

(5.82)

with coefficients a(m) independent of φ :

a(m) =∑

d≥1

1

d

∑m=m1m2

f(dm1X

T

)f(dm2X

T

)(5.83)

Since f has compact support at infinity, it follows that a(m) = 0 unless m ¿ T 2. In

the relevant range,

a(m) = O(τ(m) log T ) (5.84)

and τ(m) is the divisor function. Similarly,

∣∣∣∣∣∞∑

k=1

λφ(k)

k1/2+iTf(kY

K

)∣∣∣∣∣

2

=∑

k¿K2

λφ(k)b(k)√k

(5.85)

where

b(k) =∑

d≥1

1

d

∑

k=k1k2

(k1

k2

)iTf(dk1Y

K

)f(dk2Y

K

)(5.86)

and satisfies

b(k) = O(τ(k) log K), if k ¿ K2 (5.87)

and b(k) = 0, otherwise.

110

5.4.5 Applying the trace formula

Returning to eq. 5.81, we can now rewrite the right-hand side as

J∞ =∑

φ

αφh∆(tφ) ·[ ∞∑

m=1

λφ(m)a(m)√m

][ ∞∑

k=1

λφ(k)b(k)√k

]

=∑

m¿T 2

∑

k¿K2

a(m)b(k)√mk

·∑

φ

αφh∆(tφ)λφ(m)λφ(k) (5.88)

We discussed Kuznetsov’s trace formula in Section 5.2.2. Since h∆ satisfies the con-

ditions required by this theorem, we have

∑

φ

αφh∆(tφ)λφ(m)λφ(k) +1

π

∫ ∞

−∞

τir(m)τir(k)

|ζ(1 + ir)|2 h∆(r)dr

=δm,k

π2

∫ ∞

−∞t tanh(πt)h∆(t)dt +

∞∑c=1

S(m, k; c)

cg∆

(4π√

mk

c

)(5.89)

where τir(m) =∑

d1d2=m

(d1

d2

)iris the usual divisor function, and g∆ is the integral

Bessel transform of h∆ :

g∆(X) =2i

π

∫ ∞

−∞J2ir(X)h∆(r)

rdr

cosh πr(5.90)

By applying this formula, the quantity of 5.88 splits into three parts

J∞ = D + ND + ND′, (5.91)

where:

(a) D is the diagonal contribution

D :=1

π2

∫ ∞

−∞t tanh(πt)h∆(t)dt ·

∑

k¿K2

a(k)b(k)

k(5.92)

111

(we took into account the fact that K2 ≤ T 2)

(b) ND represents the non-diagonal sum of Kloosterman sums

ND :=∑

m¿T 2

∑

k¿K2

a(m)b(k)√mk

∞∑c=1

S(m, k; c)

cg∆

(4π√

mk

c

)(5.93)

(c) ND′ represents the non-diagonal contribution from the continuous spectrum

ND′ := − 1

π

∑

m¿T 2

∑

k¿K2

a(m)b(k)√mk

∫ ∞

−∞

τir(m)τir(k)

|ζ(1 + 2ir)|2h∆(r)dr (5.94)

5.4.6 Diagonal term

In practice, the diagonal term is easier to evaluate, and this also happens to be true

in our case. We have

D = I(h∆) ·∑

k¿K2

a(k)b(k)

k(5.95)

where I(h∆) is the integral

I(h∆) =1

π2

∫ ∞

−∞t tanh(πt)h∆(t)dt =

1

π2

∫ ∞

−∞

t2

Texp

(−π

(t− T

∆

)2)dt

=T∆

π2+

∆3

2π3T(5.96)

Hence I(h∆) = O(T∆). Since a(k), b(k) = O(τ(k) log K), we also have

∑

k¿K2

a(k)b(k)

k= O(T ε), ∀ε > 0

Setting ε = δ, we have

D = O(T 1+δ∆) (5.97)

112

5.4.7 Nondiagonal term: sum of Kloosterman sums

To analyze the non-diagonal ND, we need a result on the behavior of the g∆ in a

relevant range. First let us recall the setup. The parameter ∆ = T12+δH

12 satisfies

∆ ≤ 12T 1−δ by 5.78, while T itself is large (T > T0). g∆, the integral Bessel transform

of h∆, was defined at 5.90.

Lemma 5.6. Whenever

X ¿ T 1−δ∆

the following estimate holds

g∆(X) ¿ XT

∆4

with the implied constant depending on δ.

We give a proof of this lemma at the end of the chapter. In the proof we remark

that, when the argument X belongs to the range specified in the lemma, g∆(X) is in

fact negligible and hence the quantity ND itself is negligible.

Using the properties of ∆ (5.78), we find that the argument of g∆ in eq. 5.93

satisfies

X :=4π√

mk

c¿ TK

c≤ TK = T

32 H

12 = T 1−δ∆ (5.98)

(since K =√

TH). Therefore, the conditions of the lemma are satisfied, and we have

ND ¿∑

m¿T 2

∑

k¿K2

|a(m)b(k)|√mk

∞∑c=1

|S(m, k, c)|c

·( T

∆4·√

mk

c

)

¿ T 1+ε

∆4

∑

m¿T 2

∑

k¿K2

∞∑c=1

|S(m, k, c)|c2

(5.99)

Using Weil’s estimate for the Kloosterman sums

|S(m,n, c)| ≤ (m,n, c)12 c

12 τ(c)

113

it becomes straightforward that

∑m≤M

∑n≤N

∞∑c=1

|S(m, n, c)|c2

= O((MN)1+ε)

Based on this, we can bound the right-hand side of 5.99, up to a factor of T ε, by

T

∆4· T 2K2 =

T 4H

∆4=

T 4H

T 2+4δH2= T 2−4δH−1

Taking into account the extra factor O(T ε), we conclude that

ND = O(T 2−δH−1) (5.100)

Hence the non-diagonal ND is of a lower order of magnitude than the diagonal D.

5.4.8 Nondiagonal term: continuous spectrum

There is one more non-diagonal contribution besides the sum of Kloosterman sums,

namely

ND′ = −∑

m¿T 2

∑

k¿K2

a(m)b(k)√mk

· 1

π

∫ ∞

−∞

τir(m)τir(k)

|ζ(1 + 2ir)|2h∆(r)dr

Recalling the definition of the coefficients a(m) an b(k) (eq. 5.83 and 5.86), and

taking into account the fact that the divisor functions τir(n) satisfy the same Hecke

relations as λφ(n), we obtain the identity

ND′ = − 1

π

∫ ∞

−∞

∣∣∣∣∣∞∑

m=1

f(mT

)τir(m)√m

∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

f( kK

)τir(k)

k1/2+iT

∣∣∣∣∣

2h∆(r)dr

|ζ(1 + 2ir)|2 (5.101)

(We ignore the constants X and Y from the argument of f as they play no role in

the analysis.) Therefore ND′ ≤ 0 and hence this quantity can be ignored from 5.91,

114

without affecting the upper bound for J (see 5.76):

J ¿ J∞ ≤ D + ND

We will prove however that ND′ is O(T32+εH

12 ) in absolute value.

Since |ζ(1 + 2ir)|−2 = O((1 + |r|)ε) for any positive ε, we have

|ND′| ¿ T εQ (5.102)

where

Q =

∫ ∞

−∞

∣∣∣∣∣∞∑

m=1

f(mT

)τir(m)√m

∣∣∣∣∣

2 ∣∣∣∣∣∞∑

k=1

f( kK

)τir(k)

k1/2+iT

∣∣∣∣∣

2

h(r − T

∆

)dt

Opening the parentheses and using the Fourier transform, we have

Q = ∆∑

m,n≥1

ω(m,n)√mn

(m

n

)−iTh( ∆

2πlog(m/n)

)(5.103)

where

ω(m,n) =∑ 1

d1d2

f(d1α

T

)f(d1β

T

)f(d2γ

K

)f(d2δ

K

)(5.104)

with the summation variables satisfying

d1, d2 ≥ 1; m = ac, n = bd; ab = αβ, cd = γδ

Since f has compact support in [0,∞), we note that

ω(m,n) = 0 unless mn ¿ K2T 2

and ω(m,n) = O(T ε) in the non-trivial range. We split Q into a diagonal and a

non-diagonal term

Q = Qd + Qnd

115

with

Qd = ∆h(0)∞∑

m=1

ω(m, m)

m= O(T ε∆) (5.105)

the diagonal, and

Qnd = ∆∑

m,n≥1m6=n

ω(m,n)√mn

(m

n

)−iTh( ∆

2πlog(m/n)

)

the non-diagonal. Since h is a Schwartz function, the only non-negligible contribution

to Qnd comes from the terms satisfying ∆2π

log(m/n) ¿ T ε. Suppose n = m + l, with

l ≥ 1. This forces

∆l ≤ T εm (5.106)

The relevant range gives

Qnd = 2∆∑

m,l

ω(m,m + l)√m(m + l)

(1 +

l

m

)iTh( ∆

2πlog

(1 +

l

m

))

+ (negligible)

and the sum is over ∆T ε ≤ m ¿ KT and 1 ≤ l ≤ T εm

∆.

Therefore

Qnd ¿ ∆T ε ·∑

∆Tε≤m¿KT

∑

1≤l≤mTε

∆

1√m(m + l)

¿ ∆T ε ·KT · T ε

∆= KT 1+2ε (5.107)

since K =√

TH and ∆ = T δK. We can conclude that ND′ = O(T32+εH

12 ),∀ε > 0.

In particular, for ε = δ/2 we have

ND′ = O(T 1−δ/2∆) (5.108)

116

We recall the inequalities obtained at 5.74 and 5.81 :

S(2)H ¿ J ¿ J∞ = D + ND + ND′

and the estimates from 5.97, 5.100 and 5.108 :

D = O(T 1+δ∆), ND = O(T 2H−1), ND′ = O(T 1−δ/2∆)

Combining these, we arrive at the desired result of Proposition 5.5 :

S(2)H =

∑

H≤|tφ−T |≤2H

αφL2(

1

2, φ)

∣∣L(1

2+ iT )

∣∣2 = O(T32+2δH

12 ) (5.109)

5.4.9 Lemma on the Bessel transform

In this section we give a proof of Lemma 5.6. Let us recall the setup. T is the spectral

parameter, T > T0. δ > 0 is an arbitrarily small number, and the parameter ∆ is

chosen such that√

T ≤ ∆ ≤ T 1−δ

It is crucial that ∆ tends to ∞ together with T , at the same time being of a lower

order of magnitude. The test function h∆ that we use depends on both parameters

T and ∆, and is given by the formula

h∆(t) =1

2 tanh(πt)

t

T·(h(t− T

∆

)+ h

(t + T

∆

))

with h(t) an even, smooth function, rapidly decreasing at infinity. For convenience

we made the choice

h(t) = exp(−πt2)

117

The integral Bessel transform in Kuznetsov’s trace formula is

g∆(X) =2i

π

∫ ∞

−∞J2it(X)h∆(t)

tdt

cosh πt

Lemma 5.7. Given the above conditions, for:

0 ≤ X ≤ T 1−δ∆

we have

g∆(X) ¿ XT

∆4+ (negligible)

with the implied constant depending on δ. Here negligible means ON

(X

T N

),∀N ≥ 1 .

Proof. We proceed as in [Sa2] and [L-Y] :

g∆(X) =i

π

∫

Rth∆(t) tanh(πt)

J2it(X)− J−2it(X)

sinh πtdt

=i

π

∫

R

t2

T

h(

t+T∆

)+ h

(t−T∆

)

2 tanh πttanh(πt)


sinh πtdt

=i

π

∫

R

t2

Th(t− T

∆

)J2it(X)− J−2it(X)

sinh πtdt

=1

πT

∫

R

(t2h

(t− T

∆

))(u) cos

(X cosh(πu)

)du

=g+(X) + g−(X)

2(5.110)

where

g±(X) =1

πT

∫

R

(t2h

(t− T

∆

))(u)e±iX cosh πudu

In the previous sequence of equations we used Parseval’s identity as well as a formula

for the Fourier transform of the quotient of a Bessel function and the hyperbolic sine

[Bat, p. 59]J2it(X)− J−2it(X)

sinh(πt)

(u) = −i cos(X cosh(πu)) (5.111)

118

The Fourier transform of t2h(

t−T∆

)can be computed with a change of variable

∫ ∞

−∞t2h

(t− T

∆

)e(−ut)dt = ∆

∫ ∞

−∞h(t)(t∆ + T )2e(−u(t∆ + T ))dt

= ∆e(−uT )

∫ ∞

−∞h(t)(t∆ + T )2e(−u∆t)dt = ∆e(−uT )h1(∆u)

where for convenience we made the notation h1(t) = (t∆+T )2h(t). Note that h1 has

all the Sobolev norms bounded by T 2.

We thus obtain a first approximation toward g±(X):

g±(X) =∆

πT

∫ ∞

−∞e(−uT )h1(∆u)e±iX cosh(πu)

=1

πT

∫

Re(−u

T

∆

)e±iX(1+π2u2

2∆2 +··· )h1(u)du

=e±iX

πT

∫

Re(−u

T

∆

)e(±X

2π

(π2u2

2∆2+ · · · )

)h1(u)du

=e±iX

πT

∫

Re(−u

T

∆± πu2X

4∆2

)h1(u)du + O

(XT

∆4

)(5.112)

The error term O(

XT∆4

)is the source for the estimate of the lemma.

We still have to analyze the function

g1(X) =

∫ ∞

−∞h1(u)e

(Q(u)

)du (5.113)

where Q(u) is the quadratic form

Q(u) = ±Xu2

2∆2− u

T

∆

We will consider only the case of the + sign. (We ignore some absolute constants as

they do not affect the analysis.) The proof of Lemma 5.7 will be complete once we

prove the following

Proposition 5.8. For 0 ≤ X ≤ T 1−δ∆, g1(X) is negligible.

119

Proof. h1(t) = (t∆ + T )2h(t) =(∆2t2 + 2∆Tt + T 2

)h(t), and since our choice of

h satisfies h = h, it follows that h1(u) = (T + i∆u)2h(u), and hence

g1(X) = T 2

∫

Re(Q(u)

)h(u)du + 2iT∆

∫

Re(Q(u)

)uh(u)du + ∆2

∫

Re(Q(u)

)u2h(u)du

(5.114)

The ’leading term’ in the expression of g1 is T 2g2(X), where

g2(X) =

∫

Re(Q(u)

)h(u)du (5.115)

(All the other terms of 5.114 admit the same treatment.) We first determine the

critical point of the phase Q(u) (quadratic) :

Q′(u0) =Xu0

∆2− T

∆= 0 ⇔ u0 =

T∆

X

The condition X ≤ T 1−δ∆ implies u0 ≥ T δ, and by the stationary phase principle

the integral 5.115 is localized at the point u0 where the test function h(u), rapidly

decreasing, is essentially negligible. In what follows, we make this argument precise.

First we restrict the integration to a compact range:

g2(X) =

∫

|u|≤ 12u0

+

∫

|t|> 12u0

= I + II

Repeated integration by parts yields

I = e(Q(u)

) h(u)

2πQ′(u)

∣∣∣u= 1

2u0

u=− 12u0

−∫

|u|≤ 12u0

e(Q(u)

) d

du

( h(u)

2πiQ′(u)

)du

=

[(h + DQh + · · ·+ D

(N−1)Q h

) e(Q(u))

2πiQ′(u)

]u= 12u0

u=− 12u0

+

∫

|u|≤ 12u0

e(Q(u)

)D

(N)Q h(u)du

(5.116)

120

where D(0)Q h = h and D

(N)Q h(u) = − d

du

(DN−1

Q

2πiQ′(u)

).

Since Q′′(u) = X∆2 and Q′′′ ≡ 0 we have the following

Proposition 5.9.

D(N)Q h(u) =

1

(Q′(u))N

N∑j=0

cj,N

( X

∆2

)j h(N−j)(u)

(Q′(u))j

where cj,N are constant coefficients depending on N .

Proof: by induction.

We return now to eq. 5.116. Since Q′(u) = Xu∆2 − T

∆, we have

|u| ≤ 1

2u0 ⇒ |Q′(u)| ≥ T

∆− u0

2

X

∆2=

T

2∆.

First we show that the first term on the right-hand side is negligible:

[h + DQh + · · ·D(N−1)

Q h

2πiQ′(u)

]

u= 12u0

¿ h(12u0)

|Q′(12u0)|

+ · · ·

Since u0 ≥ T δ, this is negligible. Therefore

∫

|u|≤ 12u0

e(Q(u)

)h(u)du ∼

N∑j=0

cjN

(X

∆2

)j ∫

|u|≤ 12u0

h(N−j)(u)

(Q′(u))(N+j)du

¿N

N∑j=0

(X

∆

)j(∆

T

)N+j

¿N

N∑j=0

∆NT−N−jδ

¿(∆

T

)N

¿ T−Nδ

This is ON,δ(T−N), ∀N ≥ 1.

121

The estimate for II is immediate:

II =

∫

|u|≥ 12u0

e(Q(u)

)h(u)du ¿

∫ ∞

12u0

h(u) ¿ h(u0

2)

¿ exp(−T 2δ) ¿N,δ T−N , ∀N ≥ 1

We obtain g2(X) = O(T−N). The same analysis works for the other terms of 5.114

and hence we have

g1(X) = O(T−N)

for 0 < X ¿ T 1−δ∆. However, the function g1(X) is real analytic in X and, a priori,

g1(0) = 0. To see this, note that

g1(0) =

∫

Rh1(u)e

(− uT

∆

)du = h1

(− T

∆

)= (t∆ + T )2h(t)

∣∣∣t=−T/∆

= 0

Therefore we conclude that

g1(X) = O( X

TN

), ∀N ≥ 1

i.e. g1(X) is negligible. This completes the proof of Proposition 5.8.

Remark 5.4. It is crucial that g∆(X) → 0 as X ↓ 0 since this ensures the absolute

convergence of the infinite weighted sum of Kloosterman sums in the trace formula

of Kuznetsov∞∑

c=1

S(m,n, c)

cg∆

(4π√

mn

c

)

where X = 4π√

mnc

. This is the reason why in this context being negligible should

mean both rapidly decreasing in T and vanishing at X = 0 with order at least 1.

122

5.5 Concluding Remarks

In this chapter, we started with the weighted sum of L-functions

Disc.(∞; T/2) =∑

φ

w(tφ, T )L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (5.117)

where φ ranges over the countable family of Hecke-Maass cusp forms. First, we split

this sum (5.9) :

Disc.(∞; T/2) = Σbulk + Σtrans + Σ3

The quantity Σ3 corresponds to the range tφ−T À T ε where the weight w(tφ, T ) has

exponential decay, and hence Σ3 is negligible.

Σbulk represents the contribution to Disc.(∞; T/2) of the range tφ < P1, while

Σtrans represents the contribution of the range P1 ≤ tφ < P2, with the cut-off param-

eters given by

P1 = T − 1

4T 1−4δ and P2 = T +

1

4T 1−4δ

and δ an arbitrarily small positive number, to be chosen at the end.

The sum Σbulk was analyzed in Section 5.3, whose main result was (eq. 5.23) :

Σbulk = O(T 3δ)

The main result (eq. 5.55) of Section 5.4 stated that

Σtrans = O(T 3δ)

Therefore

Disc.(∞; T/2) = O(T 3δ)

We now let δ = ε/3. This finishes the proof of Theorem 5.1.

123

Chapter 6

Appendix

6.1 Preliminaries

6.1.1 The family sum

This chapter is concerned with an asymptotic estimate for the family sum

Disc.(∞; T/2) =∑

φ

w(tφ, T )L2(1

2, φ)

∣∣L(1

2+ iT, φ)

∣∣2 (6.1)

where the weight w(tφ, T ) has the following asymptotic formula given by eq. 5.6

w(tφ, T ) =2π2αφ

|ζ(1 + iT )|4exp Ω(tφ, T )

(1 + tφ)∏±(1 + |tφ ± T |) 1

2

1 + O

( 1

1 + |tφ − T |)

(6.2)

and Ω(tφ, T ) = π2(2T − |tφ + T | − |tφ − T |). We recall the relation with the L4 norm

of the Eisenstein series (Theorem 4.2):

∥∥EA

(1

2+ iT

)∥∥4

4≤ 2 Disc.(∞; T ) + O(log2 T ), T →∞

124

Therefore, in order to finish the proof of Theorem L4 (b) (section 2.4), we still need

the (asymptotic) estimate:

Disc.(∞; T ) = O(log2 T ) (6.3)

In what follows we outline a proof of this fact.

6.2 Bulk Range

As it was established in Chapter 4, the main contribution to the sum 6.1 comes from

the bulk range θT ≤ tφ ≤ (1−θ)T , where 0 < θ < 1 is an arbitrarily small parameter.

When tφ is in this range, the approximate functional equation (Theorem 5.3), in the

case of the automorphic L-functions L2(s, φ) and L(s + iT, φ)L(s− iT, φ), essentially

gives

∣∣L(1

2+ iT, φ)

∣∣2 =∞∑

a=1

1

a

∞∑m=1

f(ma2

T 2

)τiT (m)λφ(m)√m

+ · · · (6.4)

L2(1

2, φ) =

∞∑

b=1

1

b

∞∑m=1

f(nb2

T 2

)τ(n)λφ(n)√n

+ · · · (6.5)

with f(x) a positive function defined on [0,∞) which is constant (=1) near x = 0

and vanishes near x = ∞, satisfying the functional equation:

f(x) + f( 1

4π2x

)= 1

(We omit the dual sum from the approximate functional equation since it is of the

same size.) Here, τ(n) =∑

d|n 1 and τiT (n) =∑

d1d2=n diT1 d−iT

2 .

Also, in the bulk range, the weight satisfies

w(tφ, T ) ³ αφ

|ζ(1 + iT )|4T−2

125

Therefore, the estimate 6.3 reduces to

BT

T 2|ζ(1 + iT )|4 = O(log2 T ) (6.6)

where

BT :=∞∑

a,b=1

1

ab

∞∑m,n=1

f(ma2

T 2

)f(nb2

T 2

)τiT (m)τ(n)√mn

∑

θT≤tφ≤(1−θ)T

αφλφ(m)λφ(n) (6.7)

Test Function

We can regard the sum over φ as a sum over the entire discrete spectrum weighted

by the characteristic function χ[θT,(1−θ)T ]. This function is well approximated (in a

sense that can be made precise) by the weight

hT (t) :=1

tanh(πt)

t

Th0

( t

T

)(6.8)

where h0 is an even, smooth function of compact support in the interval θ ≤ |t| ≤ 1−θ,

of total mass 1. Technically, the function h0 should be the restriction of a holomorphic

function regular in the horizontal strip |=t| ≤ 1/2 + δ, suitable for an application of

Kuznetsov’s trace formula. We can overcome this difficulty by taking an analytic

function which vanishes to high enough order at the origin. For example, we could

start with h0(t) = exp(−πt2) and let our test function be

hT (t) =1

tanh(πt)

( t

T

)kh0

( t

T

)

with k a large enough, odd integer. We leave the details for a later occasion. The

factor tanh(πt) was inserted in the denominator in order to simplify the computation

of the Bessel transform.

In view of the above remark, from now on we shall consider BT a complete sum

126

(over all φ) weighted by hT (tφ).

6.2.1 Further reduction of the bulk sum

We apply Kuznetsov’s trace formula (5.89): BT = DT + NDT + CT , with

DT = I(hT )∞∑

a,b=1

1

ab

∞∑m=1

f(ma2

T 2

)f(mb2

T 2

)τiT (m)τ(m)

m(6.9)

I(hT ) =1

π2

∫

RthT (t) tanh(πt)dt (6.10)

and

NDT =∞∑

a,b=1

1

ab

∞∑m,n=1

f(ma2

T 2

)f(nb2

T 2

)τiT (m)τ(n)√mn

∞∑c=1

S(m,n, c)

ch+

T

(4π√

mn

c

)

(6.11)

Here

S(m,n, c) =∑

x(mod c)∗e(mx + nx

c

)(6.12)

is the Kloosterman sum, and

h+T (X) =

2i

π

∫ ∞

−∞hT (r)J2ir(X)

rdr

cosh(πr)(6.13)

is the Bessel transform.

The third term comes from the continuous spectrum:

CT = − 1

π

∞∑

a,b=1

1

ab

∞∑m,n=1

f(ma2

T 2

)f(nb2

T 2

)τiT (m)τ(n)√mn

∫ ∞

−∞

τit(m)τit(n)

|ζ(1 + 2it)|2hT (t)dt

= − 1

π

∫ ∞

−∞

( ∞∑a=1

∞∑m=1

)( ∞∑

b=1

∞∑n=1

) hT (t)dt

|ζ(1 + 2it)|2

Since the sum over a,m is an approximate functional equation for∏±,± ζ(s± it± iT ),

while the sum over n, b is an approximate functional equation for∏± ζ2(s ± it) at

127

s = 1/2, we have

CT =

∫ ∞

−∞|ζ(

1

2+ it)|4|ζ(

1

2+ i(t + T ))|2|ζ(

1

2+ i(t− T ))|2 hT (t)dt

|ζ(1 + 2it)|2 + · · · (6.14)

Applying the fourth moment and the subconvexity estimate for the Riemann zeta on

the critical line, we obtain

CT = O(T 1+2θ+ε) = O(T 4/3) (6.15)

6.3 Diagonal

We find an asymptotic formula for DT which gives the main term of BT . Eq. 6.9

gives DT = I(hT )J , with:

I(hT ) =1

π2

∫ ∞

−∞thT (t) tanh(πt)dt

J =∞∑

a,b=1

1

ab

∞∑m=1

f(ma2

T 2

)f(mb2

T 2

)τiT (m)τ(m)

m

6.3.1 The term I(hT )

The evaluation of I(hT ) is straightforward:

I(hT ) =1

π2

∫ ∞

−∞

t2

Th0

( t

T

)dt = (2π3)−1T 2 (6.16)

6.3.2 The term J

We will find an asymptotic formula for this quantity, using the following ingredients:

1. a subconvex estimate for ζ(s) on the critical line

2. the bound ζ′ζ(1 + iT ) = o(log T )

128

3. the bound ζ′′ζ

(1 + iT ) = O(log2 T )

To proceed with the evaluation of J , we introduce the Mellin transform

F (s) :=

∫ ∞

0

f(x)xs dx

x=

1

sF1(s) (6.17)

where F1(s) = − ∫∞0

f ′(x)xsdx is an entire function (f ≡ 1 near x = 0, hence

f ′(x) ≡ 0 near x = 0) of rapid decay in vertical strips. Combining the Mellin

inversion formula with the identity ([Ram])

∞∑m=1

τiT (m)τ(m)

ms+w+1=

ζ2(1 + s + w + iT )ζ2(1 + s + w − iT )

ζ(2 + 2s + 2w)

we obtain

J =1

2πi

∫

(10)

1

2πi

∫

(10)

T 2(s+w)F1(s)

s

F1(w)

w

ζ(1 + 2s)ζ(1 + 2w)

ζ(2 + 2s + 2w)

×∏±

ζ2(1 + s + w ± iT )dsdw (6.18)

We shift each line of integration at a time from <s = 10 and <w = 10 to <s = −14+ ε

and <w = −14

+ ε respectively, where ε > 0 is arbitrarily small. We pick up double

poles at s = 0, w = 0 and at s + w = ±iT ; we need to compute the corresponding

residues and estimate the shifted integrals.

First we move the w-integral to the left (while s remains on the line <s = 10) and

pick up a double pole at w = 0, coming from the factor

ζ(1 + 2w)

w=

1

2w2+

γ

w+ · · ·

129

near w = 0. The residue at w = 0 is

Resw=0 =1

2πiT 2s F1(s)

s

ζ(1 + 2s)

ζ(2 + 2s)

∏±

ζ2(1 + s± iT )

×[log T + γ +

F ′1

2F1

(0) +∑±

ζ ′

ζ(1 + s± iT )− ζ ′

ζ(2 + 2s)

]

We shall use the notation B = log T + γ +F ′12F1

(0), as we encounter this quantity in

several places. We have

J =1

2πi

∫

<s=10

1

2πi

∫

<w=− 14+ε

(Integrand)dsdw (6.19)

+1

2πi

∫

(10)

T 2s F1(s)

s

ζ(1 + 2s)

ζ(2 + 2s)

∏±

ζ2(1 + s± iT )×

×[B +

∑±

ζ ′

ζ(1 + s± iT )− ζ ′

ζ(2 + 2s)

]ds (6.20)

Here (Integrand) denotes the expression under the double integral J . Let J1 be the

double integral on the right-hand side, and J2 the single integral, so that J = J1 +J2.

Estimate for J1

By shifting the s-line of integration to <s = −14

+ ε (while w remains on the line

<w = −14+ ε), we encounter double poles at s = 0 and s = −w± iT . The residue at

s = −w± iT is O(T−N)∀N ≥ 1, since it involves factors of F1 and F ′1 evaluated high

in the vertical strip. We ignore these terms. By the residue theorem,

J1 =1

2πi

∫

(−1/4+ε)

1

2πi

∫

(−1/4+ε)

(Integrand)dsdw

+1

2πi

∫

(−1/4+ε)

T 2w F1(w)

w

ζ(1 + 2w)

ζ(2 + 2w)

∏±

ζ2(1 + w ± iT )

×[B +

∑±

ζ ′

ζ(1 + w ± iT )− ζ ′

ζ(2 + 2w)

]dw (6.21)

130

Using the subconvexity estimate (3.18) for ζ(s), we find that the first integral is

O(T−1/3); for the second integral, a further shift to the line <s = −12

+ ε yields

J1 = O(T−1/3) +1

2πi

∫

(−1/2+ε)

· · · dw = O(T−1/4)

The difference in the exponent is explained by the presence of ζ ′(12

+ it) for which

we can use the convexity estimate ζ ′(12

+ it) ¿ (1 + |t|) 14+ε. The outcome is: J1 =

O(T−1/4). (ε was absorbed in θ < 16.)

Evaluation of J2.

To evaluate J2 we also shift the line of integration to <s = −12+ ε, picking up double

poles at s = 0 and s = ±iT . Once again, the contribution from the latter poles is

negligible since it involves the rapidly decreasing function F1(s) evaluated high in the

vertical strip. We ignore these terms. By the residue theorem,

J2 =1

2πi

∫

(−1/2+ε)

T 2s F1(s)

s

ζ(1 + 2s)

ζ(2 + 2s)

∏±

ζ2(1 + s± iT )

×[B +

∑±

ζ ′

ζ(1 + s± iT )− ζ ′

ζ(2 + 2s)

]ds

+ Ress=0 + (negligible) (6.22)

For the first integral we use once again the subconvexity estimate for ζ(s) and the

convexity estimate for ζ ′(s) on the critical line. These give the upper bound T− 14 . It

follows that the residue at s = 0 (which is in fact Ress=0 Resw=0 Integrand) gives the

main term of J . This is

Residue = ζ(2)−1|ζ(1 + iT )|4Q

131

with Q given by

Q = (B + C)2 +1

2

∑±

[ζ ′′

ζ(1± iT )−

(ζ ′

ζ(1± iT )

)2]

+(ζ ′

ζ(2)

)2

− ζ ′′

ζ(2) (6.23)

with C = 2< ζ′ζ(1 + iT )− ζ′

ζ(2). Here we also took into account the fact that F1(0) =

f(0) = 1.

Evaluation of J.

Recall that J = J1 + J2. As it was already established, J1 = O(T− 14 ) and J2 =

ζ(2)−1|ζ(1 + iT )|4Q + O(T− 14 ). We now use the facts listed at the beginning of

this section. We know that ζ′ζ(1 ± iT ) = o(log T ), hence C = o(log T ). Therefore,

Q = log2 T +< ζ′′ζ

(1 + iT ) + o(log2 T ). Since ζ′′ζ

(1 + iT ) = O(log2 T ) (see section 4.4),

it follows that

J = J1 + J2 = ζ(2)−1|ζ(1 + iT )|4 (log2 T + O(log2 T )

)(6.24)

Combining the two results on I(hT ) and J , we obtain an asymptotic estimate for the

diagonal:

DT =3

π5T 2 log2 T |ζ(1 + iT )|4(1 + O(1)) (6.25)

6.4 Non-Diagonal

We first recall the expression of the non-diagonal (6.11):

NDT =∞∑

a,b=1

1

ab

∞∑m,n=1

f(ma2

T 2

)f(nb2

T 2

)τiT (m)τ(n)√mn

∞∑c=1

S(m,n, c)

ch+

T

(4π√

mn

c

)

132

In this section we show that NDT is of a lower order than DT . Essentially, we need

to find an absolute constant α > 0 such that

NDT = O(T 2−α) (6.26)

First, we need a result on the Bessel transform h+T .

6.4.1 Integral Bessel Transform

The Bessel transform h+T is given explicitly by

h+T (X) =

2i

π

∫

RhT (t)J2it(X)

tdt

cosh(πt)

=i

π

∫

R

t2

Th0

( t

T

)J2it(X)− J−2it(X)

sinh πtdt

Using the formula for the Fourier transform [Bat, p. 59]


sinh πt

(u) = −i cos

(X cosh(πu)

)

we obtain

h+T (X) =

1

π

∫

RF

(t2

Th0

( t

T

))(u) cos

(X cosh(πu)

)du

and here F stands for the Fourier transform. The Fourier transform of t2

Th0

(tT

)can

be easily computed by a change of variable:

∫ ∞

−∞

t2

Th0

( t

T

)e(−ut)dt = T 2F (

t2h0(t))(Tu) (6.27)

133

Let h1(t) = t2h0(t). This is a smooth, positive function, compactly supported away

from the origin. Therefore

h+T (X) = π−1T 2

∫

Rh1(Tu) cos

(X cosh(πu)

)du

= π−1T

∫

Rh1(u) cos

(X cosh(

πu

T))du

=g+(X) + g−(X)

2π+ O

(X

T 3

)(6.28)

and to evaluate g±(X) we use the Plancherel formula once again

g±(X) = T

∫

Re±iX(1+π2u2

2T2 )h1(u)du = Te±iX

∫

Re(± πXu2

4T 2

)h1(u)du

=

√2

π· T 2e±iX±iπ/4

√X

∫

Re(∓ t2T 2

πX

)h1(t)dt

=

√2


√X

∫

Re(∓ wT 2

πX

)h1(√

w)dw√

w

=

√2


√X

h2

(± T 2

πX

)(6.29)

where

h2(x) =

√xh0(

√x), x > 0

0, otherwise

(6.30)

Replacing h+T (x) by 1

2π(g+ + g−) in the expression of NDT introduces a total error

term (up to T ε) of:

∑

a,b¿T 2

1

ab

∑

m¿T 2/a2

∑

n¿T 2/b2

1√mn

∞∑c=1

|S(m,n, c)|c

·√

mn

c· 1

T 3

¿ T−3∑

a,b¿T 2

1

ab

∑

m¿T 2/a2

∑

n¿T 2/b2

∞∑c=1

|S(m,n, c)|c2

¿ T−3∑

a,b¿T 2

1

ab× T 2

a2

T 2

b2(Weil’s estimate for the Kloosterman sum)

¿ T

134

Therefore, NDT = 12π

(ND+ +ND−)+O(T 1+ε), where ND± has the same expression

as ND, only with g± instead of h+T . Therefore, it is sufficient to prove the estimate

6.26 for each ND± in part. In what follows, we will treat only the ”+” case, the other

case being entirely analogous.

We plug in the expression for g+ obtained at 6.29 and obtain

ND+ =eiπ/4

π√

2T 2

∞∑

a,b=1

1

ab

∞∑m,n=1

f(ma2

T 2

)f(

nb2

T 2

)τiT (m)τ(n)

(mn)3/4×

×∞∑

c=1

S(m,n, c)√c

e(2√

mn

c

)h2

( T 2c

4π2√

mn

)

We remark that h2 is a rapidly decreasing function, while f has compact support at

infinity, therefore the non-negligible contribution to ND+ comes from the range

T 2c√mn

≤ T ε

m ¿ T 2/a2, n ¿ T 2/b2

Hence 1 ≤ a, b, c ≤ T ε, where ε denotes an arbitrarily small positive number, from

now on, not always the same. It follows that in order to prove 6.26, it is enough to

prove, after opening the Kloosterman sums, the following estimate

S(c, x) = O(T−α) (6.31)

uniformly in

c ≤ T ε, x(mod c)∗, T 2−ε ≤ M ¿ T 2 (6.32)

where:

S(c, x) :=∞∑

m,n=1

H( m

M

)H

( n

M

)τiT (m)τ(n)

(mn)3/4e(mx + nx + 2

√mn

c

)h2

( T 2c√mn

)(6.33)

135

We note that the double sum has roughly M2 terms, hence the trivial bound is

O(M1/2+ε). Here H is a fixed, smooth function of compact support in (0,∞), which

replaces f after a convenient use of a partition of unity. The main tool for finding

cancellation in the sum S(c, x) is the Voronoi formula.

6.4.2 Voronoi summation formula

Let c ≥ 1 be an integer, x(mod c)∗ an invertible residue, and g a smooth function of

compact support in (0,∞). Then the following Voronoi identity holds (see [Ju]):

c

∞∑n=1

g(n)τ(n)e(nx

c

)=

∫ ∞

0

(log

( t

c

)− 2γ)g(t)dt

− 2π∞∑

n=1

τ(n)e(− nx

c

) ∫ ∞

0

Y0

(4π√

tn

c

)g(t)dt

+ 4∞∑

n=1

τ(n)e(nx

c

) ∫ ∞

0

K0

(4π√

tn

c

)g(t)dt (6.34)

Applying this formula to the n-sum in the expression of S(c, x) (eq. 6.33), we obtain

cS(c, x) = S0(c, x)− 2πS1(c, x) + 4S2(c, x), with

S0(c, x) =∞∑

m=1

H( m

M

)τiT (m)

m3/4e(mx

c

) ∫ ∞

0

H( t

M

)log

( t

e2γc

)t−3/4e

(2√

tm

c

)h2

( T 2c√mt

)dt

and

S1(c, x) =∞∑

m,n=1

H( m

M

)τiT (m)τ(n)

m3/4e(x

c(m− n)

)G1(m,n),

S2(c, x) =∞∑

m,n=1

H( m

M

)τiT (m)τ(n)

m3/4e(x

c(m + n)

)G2(m,n)

where

G1(m,n) =

∫ ∞

0

H( t

M

)t−3/4e

(2√

tm

c

)h2

( T 2c√mt

)Y0

(4π√

tn

c

)dt

and G2(m,n) has the same expression as G1(m,n), only with K0 instead of Y0.

136

6.4.3 Analysis of S0(c, x) and S2(c, x)

A change of variable transforms the integral in the expression of S0(c, x) into

2M1/4

∫ ∞

0

H(t2) log(Mt2

e2γc

)e(2√

Mm

ct)h2

(T 2ct−1

√Mm

) dt√t

Since M ≥ T 2−ε and c ≤ T ε, it follows that 2√

Mmc

≥ 2T 1−2ε, hence the above integral

is highly oscillatory. At the same time, the factor T 2c√Mm

in the argument of h2 is

O(T 2ε); integrating by parts N times shows that this integral is O(T−N/2), and hence

S0(c, x) is negligible.

In the case of S2(c, x), we use the well-known estimate for the K-Bessel function:

K0(y) ¿ y−1/2e−y, y À 1

The argument of K0 in the expression of G2(m,n) satisfies 4π√

tnc

À√

McÀ √

T ,

therefore K0

(4π√

tnc

) ¿ c1/2n−1/4 exp(−√T ), which shows that S2(c, x) has exponen-

tial decay. We can conclude that

cS(c, x) = −2πS1(c, x) + ON(T−N), ∀N ≥ 1 (6.35)

6.4.4 Analysis of S1(c, x)

Using the asymptotic formula

Y0(x) ≈√

2

πxsin(x− π/4), x À 1 (6.36)

137

in the expression of G1(m,n), we obtain after a change of variable

G1(m,n) =c1/2n−1/4

π√

2

∫ ∞

0

H(t)

te(2√

Mmt

c

)sin

(4π√

Mnt

c− π

4

)h2

( T 2c√Mmt

)dt

= (2√

2πi)−1(e−iπ/4G+

1 (m,n)− eiπ/4G−1 (m,n)

)+ (lower order)

with

G±1 (m,n) = c1/2n−1/4

∫ ∞

0

H(t)

te(2√

Mt

c(√

m±√n))h2

( T 2c√Mmt

)dt

= 2c1/2n−1/4

∫ ∞

0

H(t2)

te(2√

M(√

m±√n)

ct)h2

(T 2ct−1

√Mm

)dt

We note that G+1 (m,n) is again highly oscillatory, as the phase of the exponential

factor satisfies

2√

M(√

m +√

n)

c≥√

M

cÀ T 1−ε

(since c ≤ T ε), while the factor T 2c√Mm

in the argument of h2 is O(T ε). Repeated

integration by parts then shows that G+1 (m, n) is negligible. Therefore,

G1(m,n) = − eiπ/4

π√

2i· c1/2n−1/4

∫ ∞

0

H(t2)

te(2√

M(√

m−√n)

ct)h2

(T 2ct−1

√Mm

)dt

+ (lower order) (6.37)

We recall that S1(c, x) =∑∞

m,n=1 H( mM

) τiT (m)τ(n)

m3/4 e(

xc(m − n)

)G1(m,n), therefore m

is restricted to m ³ M . If n ≥ M1+ε the phase of the exponential factor has a large

first derivative, rendering G1(m,n) negligible. Hence the relevant range is included

in n ≤ M1+ε, and the phase φ(t) in the preceding integral satisfies

∂φ

∂t=

2√

M√m +

√n

m− n

c

138

If this is ≥ T 2+εc√Mm

, integrating by parts sufficiently many times, we find that G1(m, n)

is negligible. Therefore, the relevant range consists of

|m− n| ¿ T εc2 (6.38)

Since c ≤ T ε, this restricts our original quantity to

S1(c, x) = − eiπ/4

π√

2i· c1/2

∑

|h|≤T ε

e(xh

c

) ∑m−n=hm,n≥1

H( m

M

)τiT (m)τ(n)

m3/4n1/4Fc,h(

m

M,

n

M) (6.39)

with Fc,h(x, y) =∫∞0

H(t2)t

e(

2√x+√

yhtc

)h2

(T 2ct−1

M√

x

)dt a smooth function of two variables,

which satisfies

∂i+j

∂xi∂yjF (x, y) ¿ (h

c+ cT ε

)i+j

Since c, h have already been restricted to c, h ≤ T ε, we have O(T (i+j)ε) on the right-

hand side.

Therefore, the estimate S1(c, x) = O(T−α) reduces to finding uniform cancellation

in the smooth sum∑

m∼M

τiT (m)τ(m + h)

when the shift h is small, |h| ≤ T ε. Specifically, we need to prove

ΣM,h :=∞∑

m=1

H( m

M

)τiT (m)τ(m + h) = O(M1−δ) (6.40)

uniformly in |h| ≤ T ε, T 2−ε ≤ M ¿ T 2, for a positive constant δ > 0. This in turn

proves 6.26 with α = 2δ.

139

Case h = 0.

In this case, Ramanujan’s identity

∞∑m=1

τiT (m)τ(m)

ms=

ζ2(s + iT )ζ2(s− iT )

ζ(2s)

combined with the Mellin inversion formula yields

ΣM,0 =1

2πi

∫

(3)

ζ2(s + iT )ζ2(s− iT )

ζ(2s)M sG(s)ds

with G(s) =∫∞0

H(t)ts−1dt. Since H is smooth and of compact support, G(s) is entire

and rapidly decreasing in vertical strips. Shifting the integral to the line <s = 0, we

pick up poles at s = 1± iT , and by the theorem of residues we have

ΣM,0 =∑±

M1±iT G(1± iT )ζ2(1± 2iT )

ζ(2± iT )×

×[2γ + log M +

G′

G(1± iT ) + 2

ζ ′

ζ(1± 2iT )− 2

ζ ′

ζ(2± iT )

]

+1

2πi

∫

(1/2)

ζ2(s + iT )ζ2(s− iT )

ζ(2s)M sG(s)ds

The sum of residues is O(T−N),∀N > 1 since G and G′ are rapidly decreasing in

vertical strips. Therefore,

ΣM,0 ¿ M1/2

∫ ∞

−∞

|ζ(12

+ i(t + T ))|2|ζ(12

+ i(t− T ))|2|ζ(1 + 2it)| |G(1/2 + it)|dt

Using the subconvexity estimate ζ(12+it) = O((1+|t|)θ) and the bound |ζ(1+2it)|−1 =

O(tε) we obtain, under θ < 16,

ΣM,0 ¿ M1/2T 4θ+ε ¿ M5/6

which is in accord with 6.40

140

Case h 6= 0

The above method does not work in the case h 6= 0, simply because the Dirichlet series∑∞

n=1τiT (n)τ(n+h)

ns is no longer an Euler product. We shall use a different method for

finding cancellation which applies equally well to ΣM,0 and ΣM,h, h 6= 0.

6.5 Family Method

We define

SM(t) =∞∑

m=1

H( m

M

)τit(m)τ(m + h)

for t ∈ R, and

SM(φ) =∞∑

m=1

H( m

M

)λφ(m)τ(m + h)

for a Hecke-Maass cusp form φ. In particular, ΣM,h = SM(T ). Let ∆ = T 1−2δ a

parameter to be specified later, satisfying the preliminary condition√

T ≤ ∆ ≤ 14T .

By analogy with [Sa2], we embed our original quantity ΣM,h into a family sum,

and consider:

Σ(M) =∑

T−∆≤tφ≤T+∆

∣∣SM(φ)∣∣2 +

∫ T+∆

T−∆

∣∣SM(t)|2dt (6.41)

Conjecturally, there is square-root cancellation in the sum SM(φ), hence we expect

the average estimate

Σ(M) = O(T∆M1+ε

)(6.42)

to hold. We prove that 6.42 holds in the range 0 < δ ≤ δ0 for a particular δ0. This

yields the individual estimate SM(t) = O((T∆M)1/2+ε

), or

ΣM,h = O(M1−δ0/2)

141

which is precisely 6.40. The rest of this chapter is concerned with proving 6.42.

For clarity, we will only consider the case h = 0; the general situation |h| ≤ T ε is

completely analogous in this approach.

First, we complete the sum Σ(M) by means of the smooth weight

h∆(t) =1

2 tanh(πt)

t

T·(

exp(−π

(t− T

∆

)2)+ exp

(−π(t + T

∆

)2))

discussed in section 5.4.4. We have Σ(M) ¿ M εΣ(M), with

Σ(M) :=∑tφ≥0

αφh∆(tφ)∣∣SM(φ)

∣∣2 +1

π

∫ ∞

−∞

∣∣SM(t)∣∣2 h∆(t)dt

|ζ(1 + 2it)|2 (6.43)

By opening the parentheses and changing the order of summation, we have

Σ(M) =∞∑

m,n=1

H( m

M

)H

( n

M

)τ(m)τ(n)×

×[∑

φ

αφh∆(tφ)λφ(m)λφ(n) +1

π

∫ ∞

−∞

τit(m)τit(n)

|ζ(1 + 2it)|2h∆(t)dt

]

Kuznetsov’s trace formula (5.89) yields

Σ(M) =∞∑

m,n=1

H( m

M

)H

( n

M

)τ(m)τ(n) ·

[δm,nI(h∆) +

∞∑c=1

S(m,n, c)

cg∆

(4π√

mn

c

)]

with g∆ the Bessel transform of h∆.

6.5.1 Diagonal

The diagonal corresponds to m = n, and equals

Diag = I(h∆)∞∑

m=1

H( m

M

)τ 2(m)

142

The integral I(h∆) = 1π2

∫ ∞

−∞t tanh(πt)h∆(t)dt was already computed at 5.96; it

equals: T∆π2 + ∆3

2π3T. By the Mellin inversion formula, the inner sum equals

1

2πi

∫

(3)

ζ4(s)

ζ(2s)M sG(s) = MP3(log M) + O(M1/2+ε)

with P3 a polynomial of degree 3 with coefficients depending on H. Therefore,

Diag = O(T∆M log3 M

)(6.44)

which is in accord with 6.42.

6.5.2 Off-diagonal

The off-diagonal resulted in the expansion of Σ(M) is the weighted sum of Klooster-

man sums

Ndiag =∞∑

m,n=1

H( m

M

)H

( n

M

)τ(m)τ(n) ·

∞∑c=1

S(m,n, c)

cg∆

(4π√

mn

c

)(6.45)

As usual, this term is harder to analyze.

First, as it follows from the discussion in section 5.4.9, g∆(X) is negligible in the

range X ¿ T 1−ε∆; therefore, the only meaningful contribution to Ndiag comes from

the range√

mncÀ T 1−ε∆, or

c ≤ M

T 1−ε∆=: c0 (6.46)

It follows that

Ndiag =∑c≤c0

∑

x(mod c)∗

1

c

∞∑m,n=1

H( m

M

)H

( n

M

)τ(m)τ(n)e

(mx + nx

c

)g∆

(4π√

mn

c

)

+ (negligible) (6.47)

143

On the other hand,

g∆(X) =2i

π

∫ ∞

−∞h∆(r)J2ir(X)

rdr

cosh(πr)

Following [J-M], we ignore the range |t ± T | ≥ ∆ log T at a negligible cost. In the

remaining range, we use the asymptotic formula [J-M, 2.7]

J2ir(X) ∼ 1

π√

2Xexp

(iω(r,X) + πr − πi/4

)(r > 0)

with ω(r, x) = x(1− 2( r

x)2

), and obtain

g∆(X) ∼ 2i

π2·(e−iπ/4g+(X)− eiπ/4g−(X)

)

with g±(X) = (2X)−1/2∫|t−T |≤∆log T

rh∆(r)e±iω(r,X)dr.

Therefore, the estimate

∑c≤c0

∑

x(mod c)∗

1√c

∞∑m,n=1

H( m

M

)H

( n

M

)τ(m)τ(n)

(mn)1/4e(mx + nx

c± 1

2πω(r,

4π√

mn

c

))

= O(M1+ε) (6.48)

uniformly in r ∈ (T −∆ log T, T + ∆ log T ), certainly implies

Ndiag = O(T∆M1+ε

)

and hence would finish the proof of 6.42.

From now on, r will be fixed in the indicated range, and we consider only the ”+”

case, the other case being entirely analogous.

144

Let S be the sum from 6.48. By Voronoi formula (6.34) once again, we have

c∞∑

n=1

H( n

M

)τ(n)

n1/4e(

nx

c)e

( 1

2πω(r,

4π√

mn

c))

=

∫ ∞

0

H(t

M) log

( t

e2γc

)e( 1

2πω(r,

4π√

mt

c)) dt

t1/4(6.49)

− 2π∞∑

n=1

τ(n)e(−nx

c)

∫ ∞

0

H(t

M)Y0

(4π√

nt

c

)e( 1

2πω(r,

4π√

mt

c)) dt

t1/4(6.50)

+ 4∞∑

n=1

τ(n)e(nx

c)

∫ ∞

0

H(t

M)K0

(4π√

nt

c

)e( 1

2πω(r,

4π√

mt

c)) dt

t1/4(6.51)

Accordingly, S = S0 − 2πS1 + 4S2. As in a previous analysis, the quantities S0 and

S2 are negligible. That is because the phase of the exponential factor in 6.49 is

1

2πω(r,

4π√

mt

c

)=

2√

mt

c− r2c

4π2√

mt

whose partial derivative with respect to t is À√

Mc0À √

T . Repeated integration by

parts shows that the integral from 6.49 is O(T−N), ∀N ≥ 1; hence S0 is negligible. It

is even easier to check that S2 is negligible, since K0

(4π√

mtc

) ¿ (c√nt

)1/2exp

(− 4π√

ntc

),

which has exponential decay. Therefore,

S = −2πS1 + (negligible) (6.52)

Using (6.36) once again and a change of variable, we find that the integral from 6.50

equals (asymptotically) :

(π√

2)−1√

Mc1/2n−1/4

∫ ∞

0

H(t)√t

sin(4π

√Mnt

c− π

4

)e( 1

2πω(r,

4π√

Mmt

c

))dt

145

Then S1 splits : S1 = 12√

2πi

(e−iπ/4S+

1 − eiπ/4S−1), where

S±1 =√

M∑c≤c0

∑

x(mod c)∗

1

c

∞∑m,n=1

H( m

M

)τ(m)τ(n)

(mn)1/4e(x

c(m− n)

)B±

c (m,n) (6.53)

with

B±c (m,n) =

∫ ∞

0

H(t)√t

e(2√

Mt

c(√

m±√n)− r2ct−1/2

4π2√

Mm

)dt (6.54)

In the case of B+c , the phase φ(t) of the exponential factor satisfies

∂φ

∂t=

√M(

√m +

√n)

c· t−1/2 +

r2ct−3/2

8π2√

MmÀ M

c0

À T

Repeated integration by parts then shows that B+c (m,n) = O(T−N), ∀N > 1. There-

fore

S1 = − eiπ/4

2√

2πiS−1 + (negligible) (6.55)

Consequently, the quantity from 6.48 is S = eiπ/4√2i

S−1 + (negligible).

The sum S−1

If n ≥ M1+ε, the phase φ(t) of B−c (m,n) satisfies

∣∣∣∣∂φ

∂t

∣∣∣∣ ÀM1+ε/2

c0

in the range of integration, hence B−c (m,n) is negligible. Therefore, the relevant range

is included in m ³ M, n ≤ M1+ε. We rewrite the phase of the exponential factor as

φ(t) =2√

M√m +

√n

m− n

c· t1/2 − r2ct−1/2

4π2√

Mm

Once again, if |m− n| ≥ c2T ε, it follows that∣∣∂φ

∂t

∣∣ À cT ε/2, and repeated integration

by parts shows that B−c (m,n) is negligible. We conclude that the non-negligible

146

contribution comes from near the diagonal :

|m− n| ≤ T εc2 (6.56)

We let h := m− n, then

φ(t) =2√

M√m +

√n

h

c· t1/2 − r2ct−1/2

4π2√

Mm

We distinguish between three cases:

i) If h = 0, φ = − r2ct−1/2

4π2√

Mm, hence

∣∣∂φ∂t

∣∣ À T ε, unless c ≤ T ε.

ii) If h > 0, ∂φ∂t

=√

M√m+

√n

hct−1/2 + r2ct−3/2

8π2√

MmÀ (

hc

+ c)

which is large, unless c ≤ T ε.

iii) If h < 0, there is no extra cancellation dictated by stationary phase analysis.

Therefore,

S−1 =√

M∑c≤T ε

∑

0≤h≤T ε

S(0, h; c)

c

∑

m−n=h

H( m

M

)τ(m)τ(n)

(mn)1/4B−

c (m,n) (6.57)

+√

M∑c≤c0

∑

1≤h≤T εc2

S(0, h; c)

c

∑

n−m=h

H( m

M

)τ(m)τ(n)

(mn)1/4B−

c (m,n) (6.58)

where S(0, h; c) =∑

x(mod x)∗ e(xhc

) is the Ramanujan sum. Trivially, B−c (m,n) = O(1)

and τ(n) = O(nε), hence the entire sum from 6.57 is O(M1+ε). This yields

S−1 =√

M∑c≤c0

∑

1≤h≤T εc2

S(0, h; c)

cβc(h) + O(M1+ε) (6.59)

with βc(h) =∑∞

m=1 H( mM

) τ(m)τ(m+h)

(m(m+h))1/4 B−c (m,n).

147

6.5.3 The Additive Divisor Problem

It is a well-known fact (originally due to Ramanujan [Ram]) that

∑n≤X

τ 2(n) = xP3(log x) + O(x12+ε), ∀ε > 0

with P3 a polynomial of degree 3 of leading coefficient 1π2 . This can be proved by

means of the factorization∞∑

n=1

τ 2(n)

ns=

ζ4(s)

ζ(2s)

and the Perron formula. However, for h 6= 0, the Dirichlet series∑∞

n=1τ(n)τ(n+h)

ns is

no longer an Euler product, and the problem of finding an asymptotic formula with

good error term for the shifted divisor sum

Th(x) :=∑n≤x

τ(n)τ(n + h)

is more difficult, especially if one seeks uniformity in the parameter h. This problem

is known as the additive divisor problem, and it has a rich history. This subject was

initiated by Ingham and Estermann:

∑n≤x

τ(n)τ(n + h) ∼ 6

π2σ−1(h)x log2 x [In2] (6.60)

∑n≤x

τ(n)τ(n + h) = xPh(log x) + O(h

16 x

1112 log3 x

)[E] (6.61)

Here Ph(t) is a quadratic polynomial of degree 2 with leading coefficient 6π−2σ−1(h),

and the second equation holds uniformly in 1 ≤ h ≤√

x− 1/2.

Atkinson [A, p.185] indicates that, via Salie’s estimate on Kloosterman sums,

148

Estermann’s original method yields Th(x) = T 0h (x) + Eh(x), with

T 0h (x) =

∑0≤ν,i≤2

cνiσ(ν)−1 (h)x(log x)i (6.62)

and

Eh(x) = O(x89+εhε) uniformly in 1 ≤ h ≤

√x− 1/2 (6.63)

where cνi are absolute constants, and σ(ν)−1 (h) =

∑d|h

(log d)ν

d. This error term was

further improved by Heath Brown [HB] and Motohashi [Mot2], the former using

Weil’s estimate for the Kloosterman sum, the latter relating Kloosterman sums to

the spectrum of SL(2,Z) through Kuznetsov’s formula. We will use Motohashi’s

result [Mot2, Cor. 1]:

Eh(x) = O(x23+ε), uniformly in 1 ≤ h ≤ x

2027 (6.64)

Evaluation of βc(h)

We return now to 6.59. Integrating by parts and taking into account the fact that

ddx

B−c (x, x + h) ¿ M−1

(hc

+ c)

in the range m ³ M , we obtain

βc(h) =

∫ ∞

0

H( x

M

)B−c (x, x + h)

(x(x + h))1/4dTh(x)

=

∫ ∞

0

H( x

M

)B−c (x, x + h)

(x(x + h))1/4dT0(x, h) + O

M− 3

2 (h

c+ c)

∫ ∞

0

H(x

M)|Eh(x)|dx

=

∫ ∞

0

H( x

M

)B−c (x, x + h)

(x(x + h))1/4dT0(x, h) + O

(M− 1

2+ 2

3+ε(

h

c+ c)

)(6.65)

where in the last equation we employed 6.64. Using now 6.62 and a change of variable,

we find that βc(h) is a linear combination (modulo the error term) of terms

βν,i,jc =

√M(log M)iσ

(ν)−1 (h)

∫ ∞

0

H(x)B−

c (Mx, Mx + h)

(x(x + hM

))1/4(log x)jdx

=:√

M(log M)iσ(ν)−1 (h)Qj

c(h) (6.66)

149

say, for 0 ≤ ν, i, j ≤ 2. It follows that the sum S−1 of 6.59 is in turn a linear

combination of

Sν,i,j1 = M(log M)i

∑1≤c≤c0

∑

1≤h≤T εc2

S(0, h; c)σ(ν)−1 (h)

cQj

c(h) (6.67)

plus a total error term, generated by 6.65 :

Error = M23+ε

∑1≤c≤c0

∑

1≤h≤T εc2

|S(0, h; c)|c

(h

c+ c

)

Since S(0, h; c) =∑

l|(h,c) µ( cl)l ¿ gcd(h, c), we obtain

Error ¿ M23+ε

∑1≤c≤c0

∑

1≤h≤T εc2

(h, c)

c

(h

c+ c

)

¿ M23+εc3

0 (6.68)

We recall the definition of Qjc(h):

Qjc(h) =

∫ ∞

0

H(x)B−

c (Mx, Mx + h)

(x(x + hM

))1/4(log x)jdx

=

∫ ∞

0

∫ ∞

0

H(t)√t

H(x)(x(x + h

M))1/4

e( 2√

x +√

x + h/M

ht1/2

c+

r2ct−1/2

4π2M√

x

)(log x)jdtdx

Since h/M ¿ M−1+δ is small, the above integral is well approximated by

∫ ∞

0

∫ ∞

0

H(t)H(x)√tx

e(t1/2

√x

h

c+

t−1/2

√x

r2c

4π2M

)(log x)jdxdt

=

∫ ∞

0

∫ ∞

0

H(u

w)H(

1

uw)e

(h

cu +

r2c

4π2Mw

)2 logj( 1uw

)dudw

uw2(change of variable)

= Hj

(h

c,

r2c

4π2M

)

150

with Hj(u,w) a smooth function of compact support in (0,∞) × (0,∞). Therefore,

the sum Sν,i,j1 can be rewritten as

Sν,i,j1 = M(log M)i

∑c≤c0

∑

h≤T εc2

S(0, h; c)σ(ν)−1 (h)

cHj

(h

c,

r2c

4π2M

)

Since Hj is rapidly decreasing, the summand is negligible unless

r2c

4π2M¿ T ε (6.69)

But r2

MÀ T ε, and this forces c ≤ T ε and h ≤ T ε. Trivially, |S(0, h; c)| ≤ c and

σ(ν)−1 (h) = O(hε). Therefore Sν,i,j

1 = O(M1+ε). We finally obtain

S−1 ¿ M1+ε + M23+εc3

0. (6.70)

In view of 6.55, this completes the proof of the estimate 6.48, as long as the condition

c30 ≤ M

13 is satisfied.

Choice of ∆

Our choice of ∆ is subject to the restriction:

c30 ≤ M

13 (6.71)

By 6.46, c0 = MT 1−ε∆

, and hence the condition on c0 is equivalent to M3δ ≤ M1/3.

This leads to the optimal choice δ = 1/9 and ∆ = T 7/9, which produces

ΣM,h =∞∑

m=1

H( m

M

)τiT (m)τ(m + l) = O(M

1718

+ε)

uniformly in |l| ≤ T ε, which proves 6.26 with α = 19.

151

Remark 6.1. In fact, even Estermann’s original result is enough for proving a power

saving cancellation in the non-diagonal: eq. 6.61 yields 6.26 with α = 140

.

6.6 Conclusions

First, we reduced the family sum Disc.(∞; T/2) to a bulk sum, which we wrote with

the help of the approximate functional equation as BT

T 2|ζ(1+iT )|4 , with BT a sum of

bilinear form in Hecke coefficients. By means of the Kuznetsov’s trace formula, BT

was transformed into a sum of a diagonal and a non-diagonal component, BT =

DT + NDT . The main result of section 6.3 was an asymptotic estimate for the

diagonal, given at 6.25:

DT =3

π5T 2 log2 T |ζ(1 + iT )|4(1 + O(1)), T →∞

In section 6.4 we reduced the analysis of the non-diagonal to that of the shifted

convolution sums :∑

m∼M τiT (m)τ(m + h), with a small shift h. In section 6.5, we

used the family method to prove a power saving cancellation in these sums, as T is

large. This led to

NDT = O(T179

+ε), T →∞

We can conclude now that BT

T 2|ζ(1+iT )|4 = O(log2 T ). This implies

Disc.(∞; T/2) = O(log2 T )

which is what we set out to prove.

152

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the l4 norm of the eisenstein series

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