Sums of Fourier Coefficients of Modular Forms and the Gauss Circle Problem

Alexander Weston Walker

B.A. in Mathematics, Boston College, Chestnut Hill, MA, 2012 M.Sc. in Mathematics, Brown University, Providence, RI, 2015

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics at Brown University

Recommended for Acceptance by the Department of Mathematics Advisor: Professor Jeffrey Hoffstein

May 2018 c Copyright by Alexander Weston Walker, 2018. All rights reserved. This dissertation by Alexander Walker is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date Jeﬀrey Hoﬀstein, Advisor

Recommended to the Graduate Council

Date Maria Nastasescu, Reader

Date Michael Rosen, Reader

Approved by the Graduate Council

Date Andrew Campbell, Dean of the Graduate School Vitae

Alexander Weston Walker was born in Concord, New Hampshire on January 17, 1990 to Marilyn Walker (n´eeMcNeil) and Kenneth Walker. He grew up in Londonderry, New Hampshire and graduated from Londonderry High School in 2008. He received his B.A. in Mathematics from Boston College in 2012 and was awarded the Paul J. Sally Distinguished Alumnus Prize upon graduation.

Alexander began his graduate studies in the fall of 2012 at Brown University, where he received his M.Sc. in Mathematics in 2015. During his time at Brown, he taught undergraduate courses in calculus, linear algebra, and multi-variable calculus. In addition, he taught summer classes in number theory through the Summer@Brown program and led a readings course in analytic number theory through the math department’s Directed Reading Program.

When he’s not writing his thesis, Alexander is an active participant in the math department’s informal game theory seminar.

iv Abstract of “Sums of Fourier Coeﬃcients of Modular Forms and the Gauss Circle Problem,” by Alexander Walker, Ph.D., Brown University, May 2018.

The Gauss circle problem is a classic problem in number theory that concerns estimates for the number of lattice points contained in a circle of large radius. This question originates with Gauss, who proved that the number of lattice points can be approximated by the area of the enclosing circle. Well-supported conjectures suggest that the error of this approximation is surprisingly small. In this thesis, we investigate the Gauss circle problem and several variants by means of Dirichlet series.

We begin by studying the partial sums of Fourier coefficients of GL(2) cusp forms. These partial sums are conjectured to behave much like the error term in the Gauss circle problem, but are simpler in many analytic regards. We introduce Dirichlet series whose coefficients are the squares of the partial sums and prove that these series have meromorphic continuation to the entire complex plane. Much of this material has been introduced elsewhere, but this simplified analogy of the Gauss circle problem serves as an important foundation for later chapters.

We then turn our attention to the Gauss circle problem itself. Specifically, we address the generalized Gauss circle problem, which concerns estimates for the number of lattice points in k-dimensional spheres. Techniques developed in the previous (cusp form) case are modified and applied to understand the meromorphic behavior of the Dirichlet series associated to the second moment of the error term in the k-dimensional Gauss circle problem. Integral transforms are then applied to prove sharp and smooth second moment results for the lattice point discrepancies. Our results are particularly interesting in dimension three, where we develop the first

v power-savings error for the second moment of the lattice point discrepancy.

The conjectural bounds in the Gauss circle problem and its generalization to higher dimensions are of a fundamentally diﬀerent form. We recognize this phase change as a property of the weight of the underlying modular form and use this to motivate a new variant of the Gauss circle problem concerning twisted divisor sums that we call the Eisenstein series analogy.

Finally, we consider a variant of the Gauss circle problem that concerns the size of iterated partial sums of coeﬃcients of modular forms. We apply the theory of iterated partial sums to recover information about non-iterated partial sums and show how questions regarding iterated partial sums may be approached using Dirichlet series.

vi This thesis is dedicated to all those who gave me opportunities. With special mention to my wife, for also taking them with me.

vii Acknowledgements

This thesis would not have been possible without the support of my mathematical family. I’d like to thank Jeﬀ for encouraging the collaboration that makes this family metaphor work, and the automorphic N, for making room for one more newform.1

In particular, I’d like to thank my brothers and collaborators Tom Hulse, Chan Ieong Kuan, and David Lowry-Duda. Between our ﬁve joint papers and at least that many productive weekly meetings, you have left your ﬁngerprints on every page.

For everything he’s done for me outside of these collaborations, I’d like to give a second round of thanks to David the oﬃce-mate, David the roommate, and David the friend. You have turned me into a better mathematician and a better person.

Thank you to Doreen, for all the work you did to help me secure my current fellowship. Thank you to Audrey, for talking to the graduate school so that I never seem to have to. And thank you to Jeﬀ, Maria, Mike and Min, for serving on my committees.

Lastly, I’d like to thank my family and my wife for their endless love and support. You might not appreciate point-counting in conic sections but you do appreciate me.

1I’d also like to thank my tenth cousin nine times removed, Johann Carl Friedrich Gauss. viii Contents

1 Introduction1 1.1 The Gauss Circle Problem...... 1 1.2 The Generalized Gauss Circle Problem...... 3 1.3 A Connection to Modular Forms and the Cusp Form Analogy....6 1.4 Outline and Summary of Major Results...... 8

2 Background 12 2.1 Modular Forms...... 12 2.2 Eisenstein Series and Poincar´eSeries...... 15 2.3 Some L-functions attached to Modular Forms...... 16

3 The Cusp Form Analogy 18 3.1 An Elementary Decomposition...... 20 3.2 Analytic Properties of Wf (s)...... 23 3.2.1 Spectral Expansion of the Shifted Convolution...... 23 3.3 Meromorphic Continuation of the Shifted Convolution...... 28 3.3.1 Continuation of the Discrete Spectral Part of Zf (s, w)..... 28 3.3.2 Continuation of the Continuous Spectral Part of Zf (s, w)... 29 3.4 Polar Analysis of D(s, Sf × Sf )...... 34 3.4.1 Polar Analysis of Zf (s, w)...... 34 3.4.2 Polar Analysis of Wf (s)...... 36 3.4.3 Polar Analysis of D(s, Sf × Sf )...... 37 3.5 Arithmetic Applications...... 38

4 The Generalized Gauss Circle Problem 44 4.1 Algebraic Decompositions...... 47 4.2 Spectral Expansion of Zk(s, w)...... 50 4.2.1 Spectral Expansion of the Shifted Convolution Zk(s, w).... 51 k 2 k 4.2.2 Modifying |θ (z)| Im(z) 2 to be Square Integrable...... 52 4.2.3 Applying the Spectral Expansion of the Poincar´eSeries.... 56 4.3 Meromorphic Continuation of Zk(s, w)...... 59 4.3.1 Meromorphic Continuation of the Non-Spectral Part..... 59 4.3.2 Meromorphic Continuation of the Discrete Spectral Part... 62 4.3.3 Meromorphic Continuation of the Continuous Spectral Part. 63 4.4 Analytic Behavior of Wk(s)...... 67 ix 4.4.1 The Diagonal Part...... 67 4.4.2 The Discrete Part...... 70 4.4.3 The Continuous Part...... 70 4.4.4 The Non-Spectral Part...... 71 4.4.5 Polar Analysis of Wk(s)...... 72 4.5 Polar Analysis of D(s, Sk × Sk) and D(s, Pk × Pk)...... 74 4.5.1 Polar Analysis of D(s, Sk × Sk)...... 74 4.5.2 Polar Analysis of D(s, Pk × Pk)...... 76 4.5.3 Cancellation in the Poles of D(s, Pk × Pk)...... 78 4.6 Modiﬁcations in the Planar Case...... 79 4.6.1 Spectral Expansion of Z2(s, w)...... 80 4.6.2 Meromorphic Continuation of Z2(s, w)...... 82 4.6.3 Analytic Behavior of W2(s)...... 83 4.6.4 Polar Analysis of D(s, S2 × S2) and D(s, P2 × P2)...... 87

5 Arithmetic Results for the Generalized Gauss Circle Problem 92 5.1 The Discrete Laplace Transform...... 95 5.2 The Laplace Transform...... 96 5.3 The Discrete Second Moment...... 101 5.4 The Second Moment...... 107

6 A Brief Note on Sums of Coeﬃcients of Modular Forms 113 6.1 Sums of Coeﬃcients of Eisenstein Series...... 114 6.2 Weight One Eisenstein Series...... 115 6.3 Conjectures for the Eisenstein Series Analogy...... 117

7 Iterated Partial Sums 120 7.1 C´esaroand Riesz Means...... 120 7.2 Relation to the Gauss Circle Problem and the Cusp Form Analogy.. 122 7.3 Applications to (Non-Iterated) Partial Sums...... 124 7.4 Dirichlet Series Attached to Riesz Means...... 126 7.4.1 An Elementary Decomposition...... 127 α α 7.4.2 General Analytic Behavior of D(s, Rf × Rf )...... 128 1 1 7.4.3 Polar Analysis of D(s, Rf × Rf )...... 128

A Explicit Bounds for D(s, P3 × P3) 133 A.1 Explicit Bounds for W3(s)...... 133 A.1.1 The Diagonal Part...... 134 A.1.2 The Non-Spectral Part...... 134 A.1.3 The Discrete Part...... 135 A.1.4 The Continuous Part...... 136 A.2 Explicit Bounds for D(s, P3 × P3)...... 138

Index 140

Bibliography 142 x Chapter 1

Introduction

This thesis applies new techniques to investigate a number of variants on the classic Gauss circle problem. In this introductory chapter, we summarize the history and progress towards Gauss’ problem. We describe two variants of the circle problem and the common theme that ties them together: the partial sums of Fourier coeﬃcients of modular forms. We conclude with a summary of the main results of this thesis.

1.1 The Gauss Circle Problem

The story of the Gauss circle problem begins with an innocent question about point- counting, ﬁrst posed by Gauss in 1798:

How many integer lattice points (x, y) lie inside of the circle of radius √ R? In other words, how many integer solutions does x2 + y2 ≤ R have?

In the same work, Gauss gives a geometric proof that the number of solutions above is well-approximated by the area of the bounding circle, πR. To describe

Gauss’ bound precisely, we let S2(R) denote the number of lattice points in the circle √ of radius R, and deﬁne the lattice point discrepancy P2(R) by

S2(R) = πR + P2(R).

1 By relating the magnitude of the error term P2(R) to the area of a narrow annulus,

1/2 Gauss proved that P2(R) = O(R ). In some respects, this bound is unsurprising; indeed, by writing X S2(R) = r2(n), n≤R

in which r2(n) is the number of representations of n as a sum of two integer squares √ (ie. the number of lattice points on the circle of radius n), one may recognize Gauss’ error bound as a form of square-root cancellation, which is common in arithmetic problems.

Gauss’ bound would not be improved for over a century, until Sierpi´nski[Sie06]

1/3 proved that P2(R) = O(R ) by applying some recent ideas of Voronoi (1903). Fol- lowing Sierpi´nski’sresult mathematicians were naturally led to the following question, which might be called the actual “problem” underlying the Gauss circle problem:

α+ Question. What is the smallest exponent α for which P2(R) = O(R ) for all > 0?

It is widely conjectured that the optimal exponent in the Gauss circle problem

1 1 is α = 4 , despite the fact that any progress past Sierpi´nski’s α ≤ 3 bound has been hard-fought. Incremental improvements over the last century by the likes of Littlewood [LW24], van der Corput [vdCN28], Iwaniec–Mozzochi [IM88], and Huxley

131 has managed to show only that α ≤ 416 ≈ 0.3149 [Hux03]. A recent preprint due to 517 Bourgain and Watt [BW17] claims the marginal improvement α ≤ 1648 ≈ 0.31371.

On the other hand, a classic result due to Hardy [Har15] has already established

1 that the conjectured lower bound α ≥ 4 must hold. A second proof of Hardy’s lower bound was given seven years later as a consequence of the following mean-square theorem of Cram´er.

2 Theorem ([Cra22]). There exists a constant C > 0 such that

Z R 2 3/2 5/4 |P2(t)| dt = CR + O(R ). 0

1 Thus α = 4 represents not only a lower bound but the true average order of the exponent in the error bound. This is compelling evidence in support of the conjecture

1 α = 4 , which is often called the Gauss Circle Problem in the strict sense.

1.2 The Generalized Gauss Circle Problem

Of the circle problem’s many variants, none is more honest than the extension from point-counting in circles to point-counting in spheres of higher dimension.

Extending notation from the previous section, we let Sk(R) denote the number of √ lattice points in the k-dimensional sphere of radius R. A direct generalization of

k/2 Gauss’ proof shows that Sk(R) ∼ vkR as R → ∞, in which vk is the volume of the k-dimensional ball of radius one. As in dimension two, we quantify this approximation by deﬁning an error term Pk(R), such that

k Sk(R) = vkR 2 + Pk(R).

Written this way, the generalized Gauss circle problem asks the following:

αk+ Question. What is the least exponent αk for which Pk(R) = O(R ) for all > 0?

Gauss’ geometric bound readily extends to general dimension k > 1 to show that

αk ≤ (k − 1)/2. Known improvements verify that additional cancellation occurs in the error term, and partial results (Ω±-results and mean-square estimates, e.g.) point

3 towards the conjecture

  k  2 − 1, k > 2 αk = (1.1)  1  4 , k = 2.

Interestingly, this suggests that the behavior of the two-dimensional Gauss circle problem is fundamentally diﬀerent than the corresponding case in higher dimensions.

It does not appear that mathematicians gave serious consideration to the generalized Gauss circle problem until after Sierpi´nski’sseminal improvement in the planar

3 case. The ﬁrst result in higher dimensions is due to Landau [Lan15] and gives α3 ≤ 4

by extending Sierpi´nski’smethod to dimension three. Since then, bounds on P3(R)

2 have been the subject of many author’s works, and include α3 ≤ 3 (due to [Vin63]), 29 21 α3 ≤ 44 (due to [CI95]), and the current estimate α3 ≤ 32 ≈ 0.6563 (due to [HB99]).

The history of the mean-square estimate for the lattice point discrepancy in dimension three is far easier to state – it begins with a result due to Jarnik [Jar40] giving

Z R 2 2 2p P3(t) dt = CR log R + O(R log R), (1.2) 0

and ends with a recent result due to Lau [Lau99] which reduces the error to O(R2). Weak results such as these underline a sentiment of the authors of the excellent survey article [IKKN06], who write

“As most experts agree, in our familiar three-dimensional Euclidean space the analogue of the Gaussian problem is the most diﬃcult and enigmatic.”

In sharp contrast, the generalized Gauss circle problem in dimensions four and above is almost entirely understood. In dimension four, the conjectured error bound

4 for P4(R) follows from the closed-form

X m S (R) = 1 + 8 σ(m) − 32 σ = v R2 + O(R log R), 4 4 4 m≤R

in which σ(m) is the sum-of-divisors function for integer m and is 0 otherwise by slight abuse of notation. In dimension k ≥ 5, uniform application of the k = 4 bound and the recursive convolution formula

√ R X 2 Sk(R) = Sk−1(R) + 2 Sk−1(R − m ) m=1

k/2−1 suﬃce to give Pk(R) = O(R log R) by induction. Of course, this implies that the generalized Gauss circle problem is known (and true!) in dimensions four and above. (More is known than has been written. For example, it is known that the log-powers are not needed in dimension k ≥ 5.)

Our understanding of the generalized Gauss circle problem in dimensions k ≥ 4 highlights how much harder the problem becomes in dimensions two and three. Yet this is not to say that all is known about these higher-dimensional cases. As one example, we present the mean-square result for dimension four, which satisﬁes

Z R 2 3 5 P4(t) dt = CR + O(R 2 log R) 0

following a result of Jarnik [Jar40]. This error term is almost surely too large and could potentially be reduced to O(R2+) for any > 0 by introducing a secondary main term of size R5/2 that has yet to be isolated.

5 More information on the generalized Gauss circle problem and its relation to the Dirichlet divisor problem and point-counting problems in generic convex bodies can be found in the survey article [IKKN06].

1.3 A Connection to Modular Forms and the Cusp

Form Analogy

Analytic number theory is guided by the principle that arithmetic functions can be understood by the meromorphic properties of their generating functions. For this reason, many attacks on the generalized Gauss circle problem begin with the identity

X Sk(n) = rk(m), m≤n

in which rk(n) is the number of representations of n as a sum of k integer squares.

This idea is particularly fruitful because the generating function of the rk(n) satisﬁes

k X 2πinz X 2πinz k rk(n)e = r1(n)e = θ(z) , n∈Z n∈Z in which θ(z) is the classic theta function, ﬁrst introduced by Jacobi (1829). In modern terminology, we describe θ(z) as a modular form of weight 1/2 and level 4. Casting θ(z) in these terms suggests that the generalized Gauss circle problem is but one of a myriad of problems regarding the error terms in sums of Fourier coeﬃcients of modular forms. In particular, it may be that progress towards the Gauss circle problem could be motivated by progress on “simpler” analogues of θ(z).

Shortly after Sierpi´nskiinitiated the modern crusade on the Gauss circle problem, Ramanujan (1916) began investigating what would later become one of the ﬁrst cusp

6 forms, the modular discriminant. Deﬁned as the inﬁnite product

Y X ∆(z) := e2πiz (1 − e2πinz)24 = τ(n)e2πinz, n≥1 n≥1 the modular discriminant satisﬁes a family of functional equations related to the action of the modular group on the upper half plane. Ramanujan made several claims about the coeﬃcients τ(n) in 1916, including the claim that τ(n) n11/2+. This conjecture became known as the Ramanujan Conjecture (and later, as a part of the Ramanujan–Petersson Conjecture) until it was settled in 1974 following Deligne’s resolution of the Weil Conjectures.

Inspired by the Gauss circle problem analogy, one might hope that the partial sums of τ(n) experience greater-than-squareroot-cancellation, or even that

X 11 + 1 + S∆(X) := τ(n) X 2 4 . (1.3) n≤X for each > 0. And indeed, this bound was realized on average in mean-square by the work of Chandrasekharan–Narasimhan (1962-64), who proved that

Z X 2 11+ 3 12+ S∆(t) dt = CX 2 + O(X ) (1.4) 0 for all > 0.

Work by Hecke and many others has contextualized the properties of ∆(z) into the general theory of cusp forms, and the aforementioned results of Chandrasekharan– Narasimhan prove that extraordinary cancellation occurs in the partial sums of Fourier coeﬃcients of cusp forms, at least on average.

7 However, like the original Gauss circle problem, this cusp form analogy remains open. State-of-the-art is essentially due to Jutila and Hafner–Ivi´c,who establish that

1 summation increases the exponent of the error by at most 3 over the size of a cusp form’s individual coeﬃcients. Thus progress in the cusp form analogy matches that of the Gauss circle problem, circa 1906! The lack of further improvements could be due to greater abstraction, a lack of study, or both.

1.4 Outline and Summary of Major Results

Chapter2: Background Material

Chapter2 introduces many of the automorphic forms that we require in this thesis. We present the basic properties of these forms and their associated L-functions, directing the reader to in-depth references where appropriate.

Chapter3: The Cusp Form Analogy

Chapter3 discusses a variant of the Gauss circle problem dating back to the work of Jutila [Jut87] and Hafner–Ivi´c[HI89] which we call the cusp form analogy. To attack this problem, we introduce the Dirichlet series

2 X |Sf (n)| D(s, S × S ) := , f f ns+k−1 n≥1

in which Sf (n) denotes the sum of the ﬁrst n Fourier coeﬃcients of the cusp form f(z).

We prove that the Dirichlet series D(s, Sf × Sf ) has a meromorphic continuation to the entire complex plane. In Theorem 3.4.2, we take this analysis a step further to

classify rightmost poles and residues of D(s, Sf × Sf ). This information is used for

8 arithmetic applications in §3.5. For example, we prove in Theorem 3.5.4 that

2 3 X Sf (n) −n/X L( 2 , f × f) 3 3 1 + e = Γ( )X 2 + O X 2 nk−1 4π2ζ(3) 2 n≥1 for all > 0, where k is the weight of f(z).

The material in this chapter closely follows the paper [HKLW17c]. However, as the main purpose of this chapter is to introduce important ideas for later use in our attack on the Gauss circle problem, the scope of our results will be artiﬁcially limited.

Chapters4–5: The Generalized Gauss Circle Problem

Chapters4–5 present a novel attack on the generalized Gauss circle problem through the study of the Dirichlet series

2 2 X Sk(n) X Pk(n) D(s, S × S ) = ,D(s, P × P ) = (1.5) k k ns+k k k ns+k−2 n≥1 n≥1

and their meromorphic properties. Here, Sk(n) denotes the number of lattice points √ in the k-ball of radius n and Pk(n) denotes the error term in the approximation

k/2 Sk(n) = vkn + Pk(n), where vk is the volume of the unit k-sphere.

Our treatment of the two Dirichlet series in (1.5) is inspired by the investigation into D(s, Sf × Sf ) conducted in Chapter3. We prove that the series in (1.5) have meromorphic continuations and classify their rightmost poles, at ﬁrst for k ≥ 3 in §4.5 and then for k = 2 in §4.6.4.

Arithmetic applications are delayed until Chapter5, where we prove a variety of second moment results for the lattice point discrepancy Pk(n). One of these results

9 2 is Theorem 5.2.5, which proves new bounds for the Laplace transform of Pk(n) . In dimension k = 2, this estimate takes the form

∞ Z 3 1 2 −t/X 0 3 2 2 + P2(t) e dt = C2Γ( 2 )X − X + O X , 0 which improves a result of Ivi´c[Ivi01].

We conclude the chapter with Theorem 5.4.6, which presents asymptotics for the

(sharp and continuous) second moment of Pk(n). This result is novel in dimension k = 3, where it improves results of Jarnik [Jar40] and [Lau99] and produces, for the ﬁrst time, a power-savings error.

In AppendixA, we prove Lemma A.2.2, a technical lemma which further improves our result in the three-dimensional case. When this lemma is taken into account, our result becomes

Z X 0 0 2 2 C3 2 C3 C3 π 2 2− 2 + P3(t) dt = X log X + − − X + O X 11 . 0 2 2 4 3

Chapters4 and5 unify and further develop material which also appears in the recently submitted papers [HKLW17a] and [HKLW17b].

Chapter6: Sums of Coeﬃcients of Modular Forms

Chapter6 discusses a generalization of the cusp form analogy to sums of coeﬃcients of non-cuspidal modular forms. We show that theory is typically trivial, except in the case of modular forms of extremely low weight, and use this to explain the phase change in the generalized Gauss circle problem between dimensions two and three.

10 Based on our results we present a new variant of the Gauss circle problem for sums of Fourier coeﬃcients of weight one Eisenstein series. This Eisenstein series analogy is stated precisely and illustrated with some large-scale computation.

Chapter7: Iterated Partial Sums

Chapter7 discusses a variant of the Gauss circle problem which concerns iterated partial sums of Fourier coeﬃcients of modular forms. These iterated sums are mostly understood following work of Hardy [Har17] and Jutila [Jut87], which we adapt to suit our particular needs.

In §7.3, we demonstrate new ways in which these higher partial sums can be used to control the error terms in the Gauss circle problem and the cusp form analogy. We apply this theory in Corollary 7.3.2 to give a new proof of the result

k−1 + 1 − Sf (X) = Ω(X 2 4 ), which is conjecturally sharp (up to logarithmic factors, etc.).

We conclude Chapter7 by showing that the general techniques of Chapter3 can be applied to study the Dirichlet series associated to the squares of the Riesz means of the Fourier coeﬃcients of cusp forms.

11 Chapter 2

Background

In this chapter, we give a quick review of the automorphic forms we will be using frequently, including modular forms, Eisenstein series, and Poincar´eseries. We then introduce two types of L-functions associated to modular forms.

2.1 Modular Forms

Before deﬁning modular forms, let us recall that the group GL(2) acts on the upper half-plane H by means of linear fractional transformations, ie.

az + b a b γ · z := , γ = ∈ GL(2). cz + d c d

A modular form f(z) of integer weight k (and level 1) is a holomorphic function f : H → C which satisﬁes

k f(γz) = (cz + d) f(z) for all γ ∈ SL(2, Z) (2.1) and which is holomorphic at infinity. For many applications, it is useful to define modular forms which satisfy (2.1) when γ is restricted to a finite index subgroup Γ ⊂

12 SL(2, Z). One family of ﬁnite index subgroups consists of the principal congruence subgroups of level n,

1 0 Γ(n) := γ ∈ SL(2, ): γ ≡ mod n , Z 0 1

and subgroups Γ that contain Γ(n) for some n are known as congruence subgroups.

The congruence subgroup that we use most frequently in this thesis is Γ0(4), in which

a b Γ (n) := ∈ SL(2, ): c ≡ 0 mod n . 0 c d Z

We restrict our attention to modular forms defined on congruence subgroups in this thesis. In this case, holomorphicity at infinity implies that f(z) has a Fourier expansion at infinity, which we may write in the form

X f(z) = a(n)e(nz), e(z) := e2πiz. n≥0

If f(z) is chosen to be an eigenfunction for each of the Hecke operators, then the coefficients of a(n) are multiplicative (though not completely multiplicative). For this reason as well as for numerous connections to point-counting on curves over finite fields, these coefficients are of great arithmetic interest. Following Deligne’s resolution of the Weil Conjectures, we have a(n) d(n)n(k−1)/2 [Del74].

The hyperbolic manifold deﬁned by the quotient Γ\H has a cusp at inﬁnity and in general has many inequivalent cusps. A cusp form is a modular form on Γ that

vanishes at each cusp of Γ\H.

To each cusp a of Γ\H, let Γa denote the stabilizer of the cusp a under the group

action of Γ. A scaling matrix σa ∈ PSL2(R) is any matrix which satisﬁes σa∞ = a and

13 ∼ a b induces the isomorphism Γa = Γ∞ via conjugation. For each γ = ( c d ) ∈ PSL2(R), we deﬁne the slash operator of weight k, denoted |γ, as the operator which maps

−k f : H → C to the function f|γ(z) := (cz + d) f(γz). Thus cusp forms are exactly those modular forms for which f|σa (z) vanishes as z → ∞ for each cusp a.

The theory of modular forms has been generalized to include modular forms with character, which are holomorphic functions f : H → C satisfying the “twisted” functional equation

f(γz) = χ(d)(cz + d)kf(z) for all γ ∈ Γ, where χ is a Dirichlet character and Γ is a congruence subgroup. A classic example in this theory is the function

2 2 X 2πin2z X 2πinz θ (z) = e = r2(n)e , n∈Z n≥0

−1 which transforms as a modular form of weight 1 on Γ0(4) with character χ = · , as can be seen by Poisson summation. Thus the so-called j-invariant,

j(γ, z) := θ(γz)/θ(z), satisﬁes j(γ, z)2 = χ(d)(cz + d) and motivates the deﬁnition of a modular form of weight 1/2 (on Γ0(4)), as a holomorphic function f(z) for which f(γz) = j(γ, z)f(z). That is, a modular form of weight 1/2 is a function which transforms in the same way that θ(z) does.

By analogy, we deﬁne a modular form of (potentially half-integral) weight k and multiplier system j(γ, z) to be a holomorphic function f(z) which satisﬁes f(γz) =

14 j(γ, z)2kf(z). There is a great diversity in multiplier systems but this thesis will use the trivial multiplier system j(γ, z)2 = (cz + d) and the multiplier system associated to θ(z) exclusively.

2.2 Eisenstein Series and Poincar´eSeries

The modular forms deﬁned in §2.1 are holomorphic and, in particular, eigenfunctions of the hyperbolic Laplacian ∆. One of the easiest ways to construct other eigenfunctions of ∆ which are quasi-invariant under the action of a congruence subgroup Γ on

H is to average a “nice” eigenfunction over Γ.

s Applying this idea to the eigenfunction Is(z) = Im(z) produces the (real analytic) Eisenstein series

X E(z, s) := Im(γz)s. (2.2)

γ∈Γ∞\Γ

This series converges for any z ∈ H (uniformly in compacta) under the assump- tion Re s > 1, which implies that E(z, s) is real analytic in z. The Eisenstein series is also holomorphic in s, and may be extended past Re s > 1 via analytic continuation.

When Γ is the full modular group, the Fourier expansion of the Eisenstein series takes the form (see [Gol15, Theorem 3.1.8], eg.)

√ π Γ(s − 1 )ζ(2s − 1) E(z, s) = ys + 2 y1−s Γ(s)ζ(2s) s√ (2.3) 2π y X 1 s− 2 + σ1−2s(n)|n| Ks− 1 (2π|n|y)e(nx), Γ(s)ζ(2s) 2 n6=0

15 in which σν(n) is the divisor function and Kν(z) is the K-Bessel function.

For each integer h ≥ 1, we likewise consider the function Is(z)e(hz), which gener- ates the Poincar´eseries

X s Ph(z, s) = Im(γz) e(γ · hz).

γ∈Γ∞\Γ

Again, this series is real analytic in z and holomorphic in s, at ﬁrst for Re s > 1 and then extended by analytic continuation.

2.3 Some L-functions attached to Modular Forms

Let f(z) be a modular form of weight k, with Fourier expansion f(z) = P a(n)e(nz). We deﬁne the L-function associated to f(z), written L(s, f), as the Dirichlet series

X a(n) L(s, f) := . s+ k−1 n≥1 n 2

k−1 When f(z) is a cusp form, Deligne’s bound a(n) d(n)n 2 implies that L(s, f) converges absolutely in Re s > 1. (Our deﬁnition of L(s, f) is sometimes called the analytic normalization of L(s, f) for this reason.)

This formal transformation f(z) 7→ L(s, f) can be interpreted analytically as well. Speciﬁcally, since the Mellin transform of the exponential is a gamma function,

Z ∞ dy X Z ∞ dy Γ(s) X a(n) Γ(s) f(iy)ys = a(n) e−2πnyys = = L(s − k−1 , f). y y (2π)s ns (2π)s 2 0 n≥1 0 n≥1

16 In this way, modularity of f(z) induces a functional equation and analytic continuation for L(s, f), which takes the particularly simple form

k−1 −s+ 2 k−1 k Λ(s, f) := (2π) Γ s − 2 L(s, f) = i Λ(1 − s, f)

when f(z) is a cusp form of level 1. For cusp forms of level N, this functional √ equation instead relates f(z) to its Fricke involution, fe(z) := ( Nz)−kf(−1/Nz).

A second class of L-functions associated to modular forms are called Rankin– Selberg convolutions. Brieﬂy, given two modular forms f(z) = P a(n)e(nz) and g(z) = P b(n)e(nz) of weight k on Γ, we deﬁne the Rankin–Selberg L-function L(s, f × g) by the Dirichlet series

L(s, f × g) X a(n)b(n) := . ζ(2s) ns+k−1 n≥1

Rankin and Selberg (independently) showed that L(s, f × g) may be recognized in terms of an inner product of fg against the Eisenstein series on Γ. Speciﬁcally, we consider the Petersson inner product

ZZ f(z)g(z) Im(z)k,E(z, s) = f(z)g(z) Im(z)kE(z, s)dµ(z), Γ\H

in which dµ(x) = dxdy/y2 is the Haar measure on H. Expanding E(z, s) via (2.2), we

change variables and tile our shifted domains into a fundamental domain for Γ∞\H. At this point, explicit calculations give

Γ(s + k − 1) X a(n)b(n) f(z)g(z) Im(z)k,E(z, s) = , (2.4) (4π)s+k−1 ns+k−1 n≥1

and functional equations for E(·, s) induce those in the Rankin–Selberg L-function.

17 Chapter 3

The Cusp Form Analogy

In this chapter we prove a few results about the partial sums of Fourier coeﬃcients of integer weight cusp forms of level one. The material in this chapter closely follows work published by the author and collaborators in [HKLW17c]. However, as the primary purpose of this chapter is to introduce important ideas through analogy, the scope of these previous results will be artiﬁcially curtailed. A more thorough treatment of the cusp form analogy can be read in [HKLW17c] or in the thesis of David Lowry-Duda [Low17].

Throughout this chapter, we ﬁx a holomorphic cusp form f(z) of positive integral weight k on Γ = SL2(Z). Then f(z) has a Fourier expansion at the inﬁnite cusp, which we write in the form

X f(z) = a(n)e(nz), n≥1 with e(z) = e2πiz. In this chapter, we consider a variant of the Gauss circle problem which concerns estimates for the partial sums of the Fourier coeﬃcients of f(z);

18 namely, for X Sf (n) := a(m). m≤n

k−1 + For context, we remark that trivial application of Deligne’s bound a(n) n 2

k−1 +1+ implies that Sf (X) X 2 . In reality, this trivial estimate is quite weak, as the

sum Sf (X) experiences greater than square-root cancellation, with the best result of its kind due to Wu and Xu [WX15]:

k−1 + 1 −0.1185 Sf (X) X 2 3 (log X) .

Following Ω±-results and the on-average results of Chandrasekharan–Narasimhan

(as described for S∆(X) in (1.4), for example) one expects the upper bound

k−1 + 1 + Sf (X) X 2 4

to hold for all > 0. As in the Gauss circle problem, this conjecture corresponds to an increase of X1/4 over the size of the sum’s largest terms. By considering this cusp

1 form analogy a variant of the Gauss circle problem, we suppose that the + 4 bound 1 here might hold for the same reasons that the + 4 bound is expected to hold in the original circle problem.

The main results in this chapter give smooth and sharp estimates for the second

moment of Sf (n), which we obtain by ﬁrst understanding the analytic properties of the Dirichlet series

2 X |Sf (n)| D(s, S × S ) := . (3.1) f f ns+k−1 n≥1

19 Unfortunately, the sharp second moment results we produce are much weaker than that which may obtained using the axiomatization of Chandrasekharan and Narasimham [CN64]. Still, the approach presented here is more illustrative (in the eyes of the author) and is without a doubt more ﬂexible. This ﬂexibility is twofold:

ﬁrstly, direct study of D(s, Sf × Sf ) allows for computations with generic cutoﬀ functions; and secondly, there exist partial sum problems for which C–N is unable to produce separation of main terms and error terms.

In the next three sections we produce a meromorphic continuation of D(s, Sf ×Sf ) by ﬁrst decomposing it into a collection of tractable components. We then use this meromorphic continuation to classify the dominant poles and residues of D(s, Sf ×Sf ) and produce arithmetic results.

3.1 An Elementary Decomposition

To understand the Dirichlet series D(s, Sf ×Sf ) in (3.1), we decompose it algebraically into a form which is more amenable to our eventual analysis. Our result requires an integral identity known as the Mellin–Barnes integral, which we give as a lemma now.

Lemma 3.1.1 (The Mellin–Barnes Integral, 6.422(3) in [GR07]). We have

Z 1 1 Γ(z)Γ(s − z) −z s = t dz, (1 + t) 2πi (σ) Γ(s) provided that 0 < σ < Re s and | arg t| < π.

Our decomposition for D(s, Sf × Sf ) follows.

Proposition 3.1.2. Let f(z) = P a(n)e(nz) be a cusp form of weight k and let P Sf (n) = m≤n a(m) denote the partial sum of the Fourier coeﬃcients of f(z). Then

20 the Dirichlet series associated to Sf (n)Sf (n) decomposes as

2 X |Sf (n)| D(s, S × S ) : = f f ns+k−1 n≥1 1 Z Γ(z)Γ(s − z + k − 1) = Wf (s) + Wf (s − z)ζ(z) dz, 2πi (σ) Γ(s + k − 1)

in which 1 < σ < Re(s − 1). Here, Wf (s) is deﬁned by

L(s, f × f) W (s) := + Z (s, 0), (3.2) f ζ(2s) f

in which L(s, f × f) denotes the Rankin-Selberg convolution of f(z) with itself and

Zf (s, w) denotes the symmetrized shifted convolution

X X a(n)a(n − h) + a(n − h)a(n) Z (s, w) := . (3.3) f ns+k−1hw n≥1 h≥1

Proof. Throughout, we adopt the convention that a(n) = 0 for n ≤ 0. Expanding

the partial sums which deﬁne Sf (n) and Sf (n), we write

X Sf (n)Sf (n) X 1 X X D(s, S × S ) = = a(n − m) a(n − h). f f ns+k−1 ns+k−1 n≥1 n≥1 m≥0 h≥0

We address the sums over m and h in three cases, depending on the relative sizes of m and h. By slight abuse of notation, we write

X 1 X X X + + a(n − m)a(n − h) ns+k−1 n≥1 m=h≥0 m>h≥0 h>m≥0

to illustrate this decomposition. In each case, a change of variables is used to replace the variable h. In the ﬁrst case, we write m = h; in the second, m = h + `; in the

21 third, h = m + `. This yields

X 1 X a(n − m)a(n − m) ns+k−1 n≥1 m≥0 X X + a(n − m)a(n − m − `) + a(n − m − `)a(n − m) . m≥0 m≥0 `≥1 `≥1

The terms with m = 0 are considered separately and contribute the term

L(s, f × f) W (s) = + Z (s, 0). f ζ(2s) f

For the terms with m ≥ 1, the change of variables n 7→ n + m yields

X X 1 X X a(n)a(n) + a(n)a(n − `) + a(n − `)a(n) . (n + m)s+k−1 n≥1 m≥1 `≥1 `≥1

The variables n and m in the denominator can be decoupled using the Mellin–Barnes integral transform from Lemma 3.1.1. This yields

1 Z X 1 X X a(n)a(n) + a(n)a(n − `) + a(n − `)a(n) 2πi ns+k−1−zmz (σ) n,m≥1 `≥1 `≥1 Γ(z)Γ(s − z + k − 1) × dz. Γ(s + k − 1)

Restricting to σ > 1 allows us to collect the m-sum into ζ(z), while the sum

over n and ` becomes Wf (s − z) when Re s is taken suﬃciently large. Simpliﬁcation completes the proof.

Remark 3.1.3. At this stage (and for quite a while hereafter), there is nothing

special about D(s, Sf × Sf ) that would not be true of the more general series

X Sf (n)Sg(n) D(s, S × S ) = , f g ns+k−1 n≥1

22 where f(z) and g(z) are potentially distinct cusp forms of weight k. Indeed, these results are presented in greater generality in the paper [HKLW17c] and in David Lowry-Duda’s thesis [Low17]. We have chosen to present only the case f = g because the point-counting problems in later chapters require only this case.

3.2 Analytic Properties of Wf (s)

Through Proposition 3.1.2, one may understand D(s, Sf × Sf ) by analyzing the sim-

pler kernel function Wf (s) deﬁned in (3.2). The crux of this task lies in understanding

the oﬀ-diagonal term, the shifted convolution Zf (s, w) introduced in (3.3). Since the essential properties of the diagonal term are classically known, we concentrate on the oﬀ-diagonal here.

The main objective of this section and the next is the development of the mero-

morphic continuation of Wf (s) to the entire complex plane. Again, the diﬃculty

comes from the oﬀ-diagonal term Zf (s, w), which we treat using the theory of spectral expansion.

3.2.1 Spectral Expansion of the Shifted Convolution

Just as the diagonal term may be recognized as a Petersson inner product against an Eisenstein series, so too may the oﬀ-diagonal term be recognized in terms of inner products against Poincar´eseries. In this section, we study these inner products with

the eventual goal of constructing a meromorphic continuation for Zf (s, w).

Let T−1 denote the Hecke operator given by T (f(x + iy)) = f(−x + iy) and deﬁne

k 2 Vf (z) := Im(z) (ff + T−1(ff)). Since Vf (z) is an L -function on Γ\H, we may

compute the Petersson inner product of Vf (z) against the Poincar´eseries Ph(z, s). A

23 standard unfolding argument shows that

Z ∞Z 1 s+k dxdy hVf ,Ph(·, s)i = y f(x + iy)f(x + iy) + f(−x + iy)f(−x + iy) e(hz) 2 0 0 y X Z ∞ dy = a(n)a(n − h) + a(n − h)a(n) ys+k−1e−4πny . y n≥1 0

Evaluating the y-integral then gives

Γ(s + k − 1) hV ,P (·, s)i = D (s; h), f h (4π)s+k−1 f

in which Df (s; h) denotes the single-sum shifted convolution

X a(n)a(n − h) + a(n − h)a(n) D (s; h) := , f ns+k−1 n≥1

as deﬁned in [HHR16]. Then, to recover the doubly-summed shifted convolution

w Zf (s, w), we divide Df (s; h) by h and sum over all h ≥ 1; this gives

s+k−1 X X Df (s; h) (4π) X hVf ,Ph(·, s)i Z (s, w) = = , (3.4) f hw Γ(s + k − 1) hw n≥1 h≥1 h≥1

provided that Re s and Re w are both suﬃciently large.

The inner product hVf ,Phi in (3.4) may be understood by replacing Ph(z, s) with

its spectral expansion. To this end, let {µj} denote an orthonormal basis of Maass

1 Hecke-eigenforms with associated types 2 + itj. These Maass forms admit Fourier expansions in terms of the classical K-Bessel function; namely,

1 X 2 2πinx µj(z) = y ρj(n)Kitj (2π|n|y)e . (3.5) n6=0

24 From these coeﬃcients ρj(n) we deﬁne the associated L-functions

X ρj(n) L(s, µ ) = ρ (1)−1 . j j ns n≥1

Following Selberg’s Spectral Theorem (as presented in [IK04, Theorem 15.5]), the spectral expansion of the Poincar´eseries may be written

X Ph(z, s) = hPh(·, s), µjiµj(z) (3.6) j Z ∞ 1 1 1 + hPh(·, s),E(·, 2 + it)iE(z, 2 + it) dt. (3.7) 4π −∞

The terms at right in (3.6) and (3.7) will be called the discrete spectral contribution and the continuous spectral contribution, respectively.

To work explicitly, we compute the inner products of Maass forms and Eisenstein series against the Poincar´eseries using standard unfolding arguments. From the Fourier series (3.5) and (2.3) and the integral identity [GR07, §6.621(3)], we have

√ 1 1 π ρj(h) Γ(s − 2 + itj)Γ(s − 2 − itj) hPh(·, s), µji = , (3.8) s− 1 (4πh) 2 Γ(s) w+ 1 w− 1 2π 2 h 2 σ1−2w(h) Γ(s + w − 1)Γ(s − w) hPh(·, s),E(·, w)i = . (3.9) s− 1 ζ(2w)(4πh) 2 Γ(s)Γ(w)

1 1 1 Note that these two formulas require Re s > 2 + | Im tj| and Re s > 2 + | Re w − 2 |, 1 respectively. For w = 2 + it and t real, (3.9) specializes to

√ 1 1 1 2 πσ2it(h) Γ(s − 2 + it)Γ(s − 2 − it) hPh(·, s),E(·, + it)i = , (3.10) 2 s− 1 it ∗ Γ(s)(4πh) 2 h ζ (1 − 2it) in which ζ∗(2s) = π−sΓ(s)ζ(2s) is the completed Riemann zeta function.

25 Substitution of (3.8) into (3.6) shows that the discrete spectral contribution (3.6) may be written

√ π X 1 1 1 ρj(h)Γ(s − 2 + itj)Γ(s − 2 − itj)µj(z). s− 2 (4πh) Γ(s) j

1 This series is analytic (as a function of s) in the right half-plane Re s > 2 + θSEC,

where θSEC := supj | Im(tj)|. In general, the claim that θSEC = 0 (for any congruence subgroup) is known as Selberg’s Eigenvalue Conjecture. Fortunately, partial progress

towards this conjecture due to [Hux85] has established that θSEC = 0 for Γ = SL2(Z). 1 Thus the discrete spectral contribution is in fact analytic for Re s > 2 .

Likewise, substituting (3.10) into (3.7) shows that the continuous spectral term takes the form

√ π Z ∞ σ (h) Γ(s − 1 + it)Γ(s − 1 − it) 2it 2 2 E(z, 1 + it) dt, s− 1 it ∗ 2 2π(4πh) 2 −∞ h ζ (1 − 2it)Γ(s)

1 which is again analytic in the right half-plane Re s > 2 .

Substituting these expressions for the discrete and continuous spectral contributions into (3.4) and performing the sum over h ≥ 1 gives a spectral expansion of

Zf (s, w), which we record now.

26 P Proposition 3.2.1. Let f(z) = a(ne(nz) be a weight k cusp form on SL2(Z).

Then the shifted convolution sum Zf (s, w) may be written

X X a(n)a(n − h) + a(n − h)a(n) Z (s, w) = f ns+k−1hw n≥1 h≥1 (4π)k X = ρ (1)G(s, it )L(s + w − 1 , µ )hV , µ i (3.11) 2 j j 2 j f j j k Z (4π) 1 + G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)i dz (3.12) 4πi (0)

1 3 when Re s > 2 and Re(s + w) > 2 , where G(s, z) and ζ(s, w, z) are the collected gamma and zeta factors of the discrete and continuous spectra,

Γ(s − 1 + z)Γ(s − 1 − z) G(s, z) = 2 2 , (3.13) Γ(s)Γ(s + k − 1) ζ(s + w − 1 + z)ζ(s + w − 1 − z) ζ(s, w, z) = 2 2 . (3.14) ζ∗(1 + 2z)

We refer to lines (3.11) and (3.12) as the discrete spectral part and the continuous

spectral part of Zf (s, w), respectively.

Proof. The formal veriﬁcation of Proposition 3.2.1 follows at once from the Dirichlet

P s series identity ζ(s)ζ(s − ν) = σν(n)/n and simpliﬁcation. However, some care is needed to guarantee that the sums and integrals written above converge where stated.

We note by Stirling’s approximation that G(s, z) experiences exponential decay in | Im z| (for ﬁxed s) and that G(s, z) grows at most polynomially in | Im s| away from poles within each vertical strip. Polynomial bounds can also be obtained for the

other terms in (3.11) and (3.12). For L(s, µj) and ζ(s, w, z) this follows by functional equations and convexity bounds, while for the inner products it follows by Watson’s triple product formula (cf. [Wat08, Theorem 3]) in the discrete case and another

27 convexity argument in the continuous case. Exponential decay in | Im z| is suﬃcient to show that the sums and integrals in (3.11) and (3.12) actually converge.

Remark 3.2.2. These observations imply moreover that Zf (s, w) experiences at most

1 3 polynomial growth in vertical strips in the region where Re s > 2 and Re(s + w) > 2 .

After the full meromorphic continuation of Zf (s, w) is given, we will see that this

result actually holds for Zf (s, w) on any vertical strip in s.

3.3 Meromorphic Continuation of the Shifted

Convolution

In this section, we apply the spectral decomposition of Zf (s, w) from Proposition 3.2.1

2 to prove that Zf (s, w) has a meromorphic continuation to C . Our discussion follows

the natural decomposition of Zf (s, w) into its discrete and continuous spectral terms, which we treat separately.

3.3.1 Continuation of the Discrete Spectral Part of Zf (s, w)

The meromorphic continuation of the discrete spectral part of Zf (s, w) is induced by

the meromorphic continuations of the individual L-functions L(s, µj) that comprise

it. Since the j-sum in (3.11) converges absolutely away from poles and L(s, µj) is entire, we see that the only poles in the discrete part arise from G(s, itj).

The collection of gamma factors G(s, itj) gives rise to apparent poles when s =

1 1 2 ±itj −n for each integer n ≥ 0. However, the rightmost cluster of poles at s = 2 ±itj do not actually occur in the distinguished case w = 0. The reason for this vanishing

depends on whether the particular Maass form µj is even or odd, so we present two lemmas to make our argument.

28 Lemma 3.3.1. Fix an even Maass form µj and n ∈ Z≥0. Then L(−2n±itj, µj) = 0.

Proof. The completed L-function attached to a Maass form µj is given by

s + + it s + − it Λ (s) = π−sΓ j Γ j L(s, µ ) = (−1)Λ (1 − s), j 2 2 j j following [Gol15, §3.13], where = 0 if the Maass form is even and = 1 if it is odd.

Since the completed L-function is entire, L(−2n ± itj, µj) must be a trivial zero.

Lemma 3.3.2. Suppose that f(z) is a weight k ∈ Z cusp form. Then hVf , µji = 0

for each odd Maass form µj.

Proof. Since µj is odd we have T−1µj = −µj by deﬁnition. On the other hand,

k k T−1Vf = T−1(Im(z) (ff + T−1(ff)) = Im(z) (T−1(ff) + ff) = Vf .

Since T−1 is self-adjoint with respect to the Petersson inner product, it follows that

hVf , µji = hT−1Vf , µji = hVf ,T−1µji = −hVf , µji.

Thus hVf , µji = 0 as claimed.

We conclude from Lemmas 3.3.1 and 3.3.2 that the discrete spectral part of Zf (s, 0)

1 is analytic in the right half-plane Re s > − 2 and extends meromorphically to C with 1 potential simple poles at points of the form 2 ± itj − n, for each integer n ≥ 1.

3.3.2 Continuation of the Continuous Spectral Part of

Zf (s, w)

The meromorphic continuation of the continuous spectral part is complicated by the entanglement of several complex variables. Nevertheless, this term can be understood

29 using nothing more than delicate contour shifting and Cauchy’s residue theorem.

We recall that the contribution towards Zf (s, w) from the continuous spectrum is

k Z (4π) 1 G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz, (3.15) 4πi (0)

in which G(s, z) and ζ(s, w, z) represent collected gamma and zeta factors, as deﬁned in (3.13) and (3.14), respectively.

1 Observe that G(s, z) is analytic for Re s > 2 +| Re z| and that ζ(s, w, z) is analytic 3 for Re(s + w) > 2 + | Re z|. In particular, (3.15) is analytic in the region deﬁned by 3 1 Re(s + w) > 2 and Re s > 2 . As we expect to specialize to w = 0, we consider the 3 poles at s + w ± z = 2 as representing the dominant (with respect to s) singularities of the continuous spectrum, and we therefore treat them ﬁrst.1

To describe the meromorphic continuation of (3.15) past these apparent polar

3 3 lines, we begin by considering s + w in the domain bounded by ( 2 ) and C + 2 , in which C is a curve (to be speciﬁed further) that lies just to the right of the line (0). By shifting the contour of z-integration from (0) we will produce a meromorphic continuation of (3.15) to a slightly larger domain.

This contour shift is complicated by the presence of the term ζ∗(1 − 2z), which

1 appears in the denominator of the Fourier expansion of E(·, z + z) and threatens to place poles at the non-trivial zeros of the Riemann zeta function. Without assuming non-trivial progress towards the Riemann hypothesis, one cannot shift the z-contour by a uniform amount sideways in this step. Let us assume (by further specifying)

1In Chapter7 we consider a problem that specializes w to various negative integers. Yet even in 3 these cases, the pole at Re(s + w) = 2 remains dominant.

30 that C lies to the left of each zero of ζ(1 − 2z) and shift (0) to C.

3 This contour shift kills the symmetry of the two entangled poles at s + w ± z = 2 3 and passes by the pole at s+w = 2 +z, extracting a residue. By the residue theorem,

k Z (4π) 1 G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz 4πi (0) k Z (4π) 1 (4π)k = G(s, z)ζ(s, w, z)hVf ,E(·, − z)idz − Res GζhVf ,Ei. (3.16) 2 2 3 4πi C z=s+w− 2

3 The residue is analytic in s for s + w in the domain bounded by (1) and C + 2 . We note as well that the shifted contour integral is analytic in s + w in the region

3 3 bounded by 2 − C and 2 + C, in which −C is deﬁned as the contour obtained by reﬂecting C over the imaginary axis.

To further continue (3.15), we suppose at the start that s and w are chosen such

3 3 that s + w lies between the curves 2 − C and 2 and interpret (3.15) using (3.16). Shifting the contour of integration from C back to (0) passes the other pole, at

3 z = 2 − s − w, and we have

k Z (4π) 1 G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz 4πi C k Z (4π) 1 (4π)k = G(s, z)ζ(s, w, z)hVf ,E(·, − z)idz + Res GζhVf ,Ei. (3.17) 2 2 3 4πi (0) z= 2 −s−w

Again, this residue has an easily-understood meromorphic continuation to the complex plane. We note that the newly unshifted contour integral is analytic in the

3 1 region Re(s + w) < 2 and Re s > 2 . Thus the integral (3.15), originally deﬁned for 1 3 1 Re s > max( 2 , 2 − Re w), has a meromorphic continuation to the strip 2 < Re s <

31 3 2 − Re w given by

k Z (4π) 1 G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz 4πi (0) (4π)k (4π)k + Res GζhVf ,Ei − Res GζhVf ,Ei. (3.18) 3 3 2 z= 2 −s−w 2 z=s+w− 2

Moreover, this meromorphic continuation is an honest-to-goodness analytic continuation to the half plane Re(s) > 1. To verify this claim, we just need to understand the meromorphic properties of the pair of residues in line (3.18), which we call

+ − R 3 (s, w) and R 3 (s, w), respectively. Direct computation shows that 2 2

k + (4π) Γ(1 − w)Γ(2s − 2 + w)ζ(2s + 2w − 2) R 3 (s, w) = ∗ Vf ,E(·, s + w − 1) (3.19) 2 2Γ(s)Γ(s + k − 1)ζ (4 − 2s − 2w) k − (4π) Γ(2s − 2 + w)Γ(1 − w)ζ(2s + 2w − 2) R 3 (s, w) = − ∗ Vf ,E(·, 2 − s − w) . 2 2Γ(s)Γ(s + k − 1)ζ (2s + 2w − 2)

+ − Proposition 3.3.3. We have R 3 (s, w) = −R 3 (s, w). 2 2

Proof. Canceling like terms, it suﬃces to prove that

∗ ∗ ζ (2s + 2w − 2)hVf ,E(·, s + w − 1)i = ζ (4 − 2s − 2w)hVf ,E(·, 2 − s − w)i.

To prove this claim, recall that the completed Eisenstein series E∗(z, s), deﬁned by ζ∗(2s)E(z, s), satisﬁes the functional equation E∗(z, s) = E∗(z, 1 − s). We bring the zeta functions into the inner products to complete the Eisenstein series and relate them using the symmetry of E∗(·, s).

1 3 Thus (3.15) has a meromorphic continuation into 2 < Re s < 2 − Re w given by

k Z (4π) 1 + G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz + 2R 3 (s, w). 4πi (0) 2

32 We observe as well that the contour integral is analytic in this vertical strip and

+ that the only pole of R3/2(s, w) in this region lies at s = 1 − w/2. (This pole appears in this vertical strip if and only if Re w < 1, which is equivalent to the claim that the zeta residues were rightmost.) Again assuming that Re w < 1, this

3 gives an analytic continuation of (3.15) past Re s > 2 −Re w into Re s > 1−Re(w)/2.

We now iterate this argument to produce a full meromorphic continuation for (3.15). Fortunately, our task is simpliﬁed by the uniform nature of further polar

1 lines: these lines arise from poles in G(s, z) and take the form s ± z = 2 − j, for each integer j ≥ 0. As before, contour shifting and unshifting suﬃces to show that (3.15) has a meromorphic continuation into successively larger half-planes, obtained by accounting for additional pairs of residues.

The conclusion of this iterative argument is presented in the following proposition. To state our result precisely, we deﬁne a general form of the Kronecker delta notation,

written δ[P ], which returns 1 when the proposition P is true and otherwise returns 0.

Proposition 3.3.4. The continuous spectral term of Zf (s, w), as given by (3.15), has meromorphic continuation to C2. This continuation may be written explicitly as

k Z (4π) 1 G(s, z)ζ(s, w, z)hVf ,E(·, 2 − z)idz 4πi (0)

+ X + − + δ[Re(s+w)< 3 ]2R 3 (s, w) + R 1 (s, w) − R 1 (s, w) , 2 2 2 −j 2 −j 1 j< 2 −Re s

in which, for each j ≥ 0, we deﬁne

k ± (4π) 1 R 1 (s, w) = Res G(s, z)ζ(s, w, z) Vf ,E(·, 2 − z . 2 −j 1 2 z=±( 2 −s−j)

33 ± Remark 3.3.5. As with R3/2, we have anti-symmetry in the further pairs of residual terms. This can be seen by observing that each of the factors

ζ(s, w, z) G(s, z), , and ζ∗(1 − 2z)hV ,E(·, 1 − zi ζ∗(1 − 2z) f 2

are even functions of the variable z. (The “odd”-ness enters when taking the residue.) Thus the j-sum in Proposition 3.3.4 can be written as twice the sum of an appropriate

+ collection of residues R 1 (s, w). 2 −j

3.4 Polar Analysis of D(s, Sf × Sf )

The results of the previous section establish that the constituent parts of Wf (s) have meromorphic continuations to the entire complex plane. As a consequence, we see

that D(s, Sf × Sf ) is meromorphic in C. Yet this statement is more or less useless without a lot more work, since there happens be a large amount of polar cancellation among the various terms in our decompositions for D(s, Sf × Sf ).

The ultimate goal in this section is to classify the dominant (ie. rightmost) poles

of D(s, Sf × Sf ) in preparation for arithmetic applications. We do so in a three-step

program: a polar analysis of Zf (s, 0), then of Wf (s), and ﬁnally of D(s, Sf × Sf ).

3.4.1 Polar Analysis of Zf (s, w)

We recall that the discrete spectral part of the shifted convolution Zf (s, w) has a

1 cluster of inﬁnitely many poles on the line Re s = − 2 . For this reason, we restrict our 1 analysis to the half-plane Re s > − 2 and focus on the contribution of the continuous spectral part. Through Proposition 3.3.4 we see that the poles of the continuous

± 1 spectral part arise solely within its residual terms R . Yet in the region Re s > − 2 ,

34 + + only R 3 and R 1 have appeared. 2 2

The rightmost residual pair, specialized to the value w = 0, takes the form

k s−1 + − (4π) Γ(2s − 2)π 2R 3 (s, 0) = −2R 3 (s, 0) = Vf ,E(·, 2 − s) 2 2 Γ(s)Γ(s + k − 1)Γ(s − 1)

3 and appears once Re(s) < 2 . To identify the poles of this expression, we apply the 1 1−2z√ Gauss duplication formula, ie. Γ(z)Γ(z + 2 ) = 2 π Γ(2z), to simplify the ratio Γ(2s − 2)/Γ(s − 1). This gives

3 1 s+k− 2 + (4π) Γ(s − 2 ) 2R 3 (s, 0) = Vf ,E(·, 2 − s) 2 Γ(s)Γ(s + k − 1)

1 and indicates that the only poles in the right half-plane Re s > 2 occur at s = 1 (from 1 1 the inner product) and at s = 2 (from Γ(s − 2 )). For the pole at s = 1, we calculate

k + (4π) L(s, f × f) Res 2R 3 (s, 0) = Res Vf ,E(·, 2 − s) = − Res , (3.20) s=1 2 s=1 Γ(k) s=1 ζ(2)

in which we’ve used the inner product representation of the Rankin–Selberg L-

1 function L(s, f × f) from (2.4) to simplify. For the pole at s = 2 , we simplify and again interpret the inner product via L(s, f × f) to show that

3 1 3 k−1 k− 2 + 4 π 3 (k − 2 ) L( 2 , f × f) Res 2R 3 (s, 0) = 1 Vf ,E(·, 2 ) = 2 . (3.21) s= 1 2 4π ζ(3) 2 Γ(k − 2 )

The second residual pair enters into the meromorphic continuation of the contin-

1 uous spectral part once Re s < 2 . The contribution of this pair is

k + (4π) ζ(0)ζ(2s − 1)Γ(2s − 1) 2R 1 (s, 0) = ∗ hVf ,E(·, s)i . 2 ζ (2 − 2s)Γ(s)Γ(s + k − 1)

35 Un-completing the completed zeta function, applying the Gauss duplication formula, reﬂecting ζ(s) and E(s, w) via functional equations, and simplifying gives

+ L(s, f × f) 2R 1 (s, 0) = − . 2 ζ(2s)

This has apparent poles at s = 1 and at the zeros of ζ(2s). However, we note that

1 this term is only considered in the region Re s < 2 , so we ignore this ﬁrst case. This ± 1 1 exhausts the poles of R 1 (s, 0) in the vertical strip Re s ∈ (− 2 , 2 ). 2

We postpone a summary of these results until the end of the next section, once we have taken advantage of cancellation between 2R 1 (s, 0) and the diagonal of Wf (s). 2

3.4.2 Polar Analysis of Wf (s)

Recall that Wf (s) is deﬁned by

L(s, f × f) W (s) = + Z (s, 0). f ζ(2s) f

The ﬁrst term (representing the diagonal contribution) has a simple pole at s = 1

which cancels with the pole of Zf (s, 0) calculated in (3.20). The function Wf (s) is

1 otherwise analytic until s = 2 , where it has a simple pole with residue given by (3.21). 1 Once Re s < , the diagonal term cancels identically with 2R 1 (s, 0), which completes 2 2 our polar analysis. This discussion is summarized with the following theorem.

Theorem 3.4.1. The function Wf (s) has a meromorphic continuation to the entire

1 complex plane and is analytic in the right half-plane Re s > 2 . The unique pole of 1 1 Wf (s) in the right half-plane Re s > − 2 occurs at s = 2 and has residue

3 1 3 k−1 k− 2 + 4 π 3 (k − 2 ) L( 2 , f × f) Res 2R 3 (s, 0) = 1 Vf ,E(·, 2 ) = 2 . s= 1 2 4π ζ(3) 2 Γ(k − 2 )

36 3.4.3 Polar Analysis of D(s, Sf × Sf )

We are now ready to describe the rightmost poles of the analytic continuation of the series D(s, Sf × Sf ). Recall from Proposition 3.1.2 that D(s, Sf × Sf ) is related to

the kernel Wf (s) by the integral expression

1 Z Γ(z)Γ(s + k − 1 − z) D(s, Sf × Sf ) = Wf (s) + Wf (s − z)ζ(z) dz, 2πi (σ) Γ(s + k − 1)

in which 1 < σ < Re(s − 1).

While keeping Re s suﬃciently large, we shift the line of integration to the left. Convergence of the shifted integral is guaranteed by the exponential decay of the two gamma factors in the integrand. This shift passes a pole at z = 1 in ζ(z) and further poles from Γ(z) at non-positive integers. Shifting the line of integration to (−2 + ) and extracting residues gives

W (s − 1) W (s) D(s, S × S ) = f + f + 1 (s + k − 1)W (s + 1) f f s + k − 2 2 12 f 1 Z Γ(z)Γ(s + k − 1 − z) + Wf (s − z)ζ(z) dz. 2πi (−2+) Γ(s + k − 1)

3 The shifted integral is analytic in Re s > − 2 + . In particular, we may read oﬀ the 1 poles of D(s, Sf × Sf ) in the right half-plane Re s > 2 by considering the poles of the three extracted residues. Doing so proves the following theorem.

Theorem 3.4.2. The function D(s, Sf ×Sf ), originally deﬁned as a series for Re s >

3, has a meromorphic continuation to all of C. The unique pole of D(s, Sf × Sf ) in 1 3 the right-half plane Re s > 2 occurs at s = 2 and has residue

3 Wf (s − 1) L( 2 , f × f) Res D(s, Sf × Sf ) = Res = . 3 3 2 s= 2 s= 2 s + k − 2 4π ζ(3)

37 3.5 Arithmetic Applications

The overarching philosophy of analytic number theory suggests that results like The-

2 orem 3.4.2 can be used to study the behavior of the coeﬃcients |Sf (n)| . In this setting, the simplest tool is the Wiener-Ikehara Theorem.

P s Theorem 3.5.1 (Wiener–Ikehara). Suppose that b(n) ≥ 0 and that n≥1 b(n)/n converges to an analytic function for Re s > σ with a unique simple pole of residue c at s = σ. Then X c b(n) ∼ Xσ. σ n≤X

Proof. A proof in the case σ = 1 is given in [MV06, Corollary 8.8]. The general case is similar or obtained afterwards via Abel summation.

2 As an application of this result, we obtain a weak moment result for |Sf (n)| .

Corollary 3.5.2. We have

2 3 X |Sf (n)| L( 2 , f × f) 3 ∼ X 2 . nk−1 6π2ζ(3) n≤X

k−1+ 1 It follows that the average order of Sf (n) is X 4 , which agrees with the conjecture of Hafner and Ivi´c.

However, we remark that the Wiener-Ikehara Theorem is a very blunt tool, since it does not assume anything about the behavior of the Dirichlet series past its dominant singularity. (Corollary 3.5.2 is still weaker than what could be produced using the machinery of Chandrasekharan–Narasimhan from [CN64], for example.)

Stronger forms of Corollary 3.5.2 exist that require more information about the

Dirichlet series D(s, Sf × Sf ). This information can be described in one of two ways:

38 (a) Information about the non-dominant singularities of the Dirichlet series. Such terms have the potential to give secondary asymptotics or suggest lower bounds on the size of error terms (in the case of a cluster of inﬁnitely many poles).

(b) Information about the growth of the Dirichlet series on vertical lines. This growth behavior dictates whether or not certain contour integral representations of cutoﬀ integrals will converge and will impact the size of error terms.

In the case of D(s, Sk × Sk), good (in fact, complete!) information is known about (a) and it is (b) that represents the obstacle to our arithmetic applications. Speciﬁcally, very little is known about the growth of the discrete spectral contribution as the argument grows in vertical strips. What is known is insuﬃcient to improve Chandrasekharan-Narasimhan, although it is, in principle, enough to obtain a power- savings error term:

2 3 X Sf (n) L( 2 , f × f) 3 3 −λ ∼ X 2 + O X 2 (3.22) nk−1 6π2ζ(3) λ n≤X

for some λ > 0. The details required to state (3.22) with an explicit choice of λ > 0 are documented in [HKLW17d] and are not worth the lengthy digression.

The particular bias towards (a)-information over (b)-information suggests that our most impressive results will take advantage of cutoﬀ integral transforms that provide good damping in vertical strips. The example we present here is an integral representation of the exponential function known as the Cahen–Mellin integral,

1 Z e−y = Γ(s)y−sds, (3.23) 2πi (σ)

39 where σ > 0 and Re y > 0. (This is just the deﬁnition of the gamma function, after Mellin inversion.) The Cahen–Mellin identity implies that

∞ 2 Z ∞ 2 s X Sf (n) 1 X Sf (n) X e−n/X = Γ(s) ds nk−1 2πi nk−1 n n=1 (σ) n=1 Z 1 s = D(s, Sf × Sf )X Γ(s) ds, (3.24) 2πi (σ)

provided that σ 1 to begin. Exponential decay in vertical strips from Γ(s) justiﬁes

shifting (σ) left past the poles of D(s, Sf × Sf ) provided that D(s, Sf × Sf ) grows at

most sub-exponentially in vertical strips. In the next lemma, we prove that D(s, Sf ×

Sf ) grows at most polynomially in | Im s| in vertical strips, which is more than enough for our applications.

Theorem 3.5.3. The function D(s, Sf × Sf ) grows at most polynomially in | Im s| in vertical strips away from poles.

Proof. As a preliminary step, we prove that Wf (s) is polynomially bounded in

vertical strips. Our proof follows the decomposition of Wf (s) into diagonal, discrete, and continuous parts, and we bound each part separately.

The simplest case is the diagonal part, where the result is immediate from the Phragm´en–Lindel¨of principle and a functional equation to give bounds for L(s, f ×f) in a left half-plane.

To bound the continuous part of Wf (s) we must address the growth of the residuals R±(s, 0) as well as the integral (3.15). For the rightmost residual pair

k s−1 + − (4π) Γ(2s − 2)π R 3 (s, 0) = −R 3 (s, 0) = Vf ,E(·, 2 − s) 2 2 2Γ(s)Γ(s + k − 1)Γ(s − 1)

40 our result follows from bounds on L(s, f × f) and Stirling’s approximation, which veriﬁes that the exponential contributions of the gamma factors cancel out. Further residual pairs are treated analogously.

1 1 To bound the integral (3.15), we remark that hVf ,E(·, 2 −z)i/Γ( 2 +z) and ζ(s, z) are polynomially bounded in | Im z| and | Im s|, while

Γ(s − 1 + z)Γ(s − 1 − z) G(s, z) = 2 2 Γ(s)Γ(s + k − 1) (3.25) | Im(s − z)|Re s| Im(s + z)|Re s| Im s|Ae−π max(| Im s|,| Im z|)+π| Im s| on the line Re z = 0 for some constant A > 0, by Stirling’s approximation. Thus G(s, z) decays exponentially in | Im z| > | Im s|1+ and grows at most polynomially in | Im z| < | Im s|1+. In total, the integral contributes only polynomial growth.

Finally, we consider the discrete part of Wf (s), which can be read from (3.11).

Convexity bounds for the L-function L(s, µj), the estimate (3.25) for G(s, z), and the inequality

X 2 k k+1 1 ρj(1)h|f| Im(·) , µji T (log T ) 2 (3.26)

|tj |∼T from Proposition 4.3 of [HHR16] give polynomial growth in the discrete spectrum.

To bound D(s, Sf × Sf ), it remains to bound the Mellin–Barnes integral

1 Z Γ(z)Γ(s + k − 1 − z) Wf (s − z)ζ(z) dz. (3.27) 2πi (σ) Γ(s + k − 1)

41 To do so, we collect Wf (s) into a single Dirichlet series of the form

2 X a(n)Sf (n) + a(n)Sf (n) − |a(n)| W (s) = . f ns+k−1 n≥1

k−1 + 1 The Hafner–Ivi´cbound Sf (X) = O(X 2 3 )[HI89] implies that the series above

4 converges absolutely for Re s > 3 . In particular, Wf (s) is uniformly bounded in the 4 right half-plane Re s > 3 + .

We now shift (σ) left in (3.27) until Wf (s − z) becomes uniformly bounded on the contour. The shifted integral is then polynomially bounded and our bound for Wf (s) gives a polynomial bound for any residues we might extract.

1 Under Theorem 3.5.3, we may shift the line of integration from (σ) to ( 2 + ) in (3.24) and ensure that the shifted contour integral still converges. This shift ex- tracts a pole and we ﬁnd that

2 3 Z X Sf (n) L( , f × f) 3 1 −n/X 2 3 2 s k−1 e = 2 Γ( 2 )X + D(s, Sf × Sf )X Γ(s) ds. n 4π ζ(3) 2πi 1 n≥1 ( 2 +)

The last integral can be bounded in absolute value to produce an error term of size

1 + O(X 2 ). This produces our main smoothed result.

Theorem 3.5.4. For all > 0, we have

2 3 X Sf (n) −n/X L( 2 , f × f) 3 3 1 + e = Γ( )X 2 + O X 2 . nk−1 4π2ζ(3) 2 n≥1

Remark 3.5.5. The smoothed series at left in Theorem 3.5.4 is sometimes called the discrete Laplace transform, in analogy with the ordinary (integrated) Laplace transform. Both of these transforms will be discussed in greater detail in Chapter5.

42 It is tempting to believe that clever linear combinations of exponentially smoothed sums might be used to approximate the sharp cutoﬀ n ≤ X. This is unfortunately not true. And, while it is possible to move from sharp to smooth results, this is not done in practice because error bounds in sharp cutoﬀs are almost always strictly worse than the bounds in their smoothed analogues.

In any case, the best error bound for the sharp cutoﬀ (3.22) appears to be due to Chandrasekharan–Narasimhan [CN64], who prove that O(X) holds.

43 Chapter 4

The Generalized Gauss Circle Problem

In this chapter, we adapt and extend the techniques developed for the cusp form analogy to study the generalized Gauss circle problem, a classical problem in number theory which concerns estimates for the number of lattice points contained in a k-dimensional sphere of large radius.

k To ﬁx notation, let Sk(R) denote the number of Z -lattice points contained within √ the k-dimensional ball of radius R. Following Gauss’ geometric argument, one has

k Sk(R) = vkR 2 + Pk(R),

k−1 in which Pk(R) = O(R 2 ) is an error term and vk denotes the volume of the unit k-sphere. Roughly speaking, the generalized Gauss circle problem asks how large the error term Pk(R) might be. More precisely, it asks the following:

Question (The Generalized Gauss Circle Problem). What is the smallest αk such

αk+ that Pk(R) = O(R ) for all > 0?

44 A brief introduction to the generalized Gauss circle problem and its history can be found in §1.2. More information regarding traditional attacks on the generalized Gauss circle problem can be found in the recent survey article [IKKN06].

Our attack on the generalized Gauss circle problem begins with the expression

X Sk(R) = rk(n), n≤R

in which rk(n) is the number of representations of n as a sum of k integer squares.

In the spirit of the previous chapter, we hope to understand Sk(R) by exploiting properties of the generating function for the rk(n),

k X θ (z) = rk(n)e(nz), n∈Z which is simply the kth power of Jacobi’s theta function θ(z). Now, if θk(z) were an integral weight cusp form of level 1, the methods of the previous chapter would apply to the generalized Gauss circle problem directly, and our work here would be done.

Unfortunately, θk(z) is not a cusp form. However, it is a modular form, which gives some hope that the general techniques from the previous chapter may still apply. In this chapter, we show how to overcome the technical diﬃculties that diﬀerentiate the cusp form analogy from the generalized Gauss circle problem. In practice, this amounts to addressing the following concerns:

k (a) The function θ (z) does not transform under the full modular group SL2(Z),

but rather under the congruence subgroup Γ0(4). Since the hyperbolic surface

Γ0(4)\H has three cusps and Selberg’s Spectral Theorem includes a sum

45 indexed over the cusps of the surface, this has the eﬀect of complicating the exact form of the spectral decomposition we apply.

(b) Since θ(z) is a modular form of weight 1/2, θk(z) is a modular form of weight k/2. In particular, θk(z) is a half-integral weight modular form when k is odd. This matters surprisingly little in our analysis but will occasionally force us to prove results without taking advantage of certain closed forms (such as Euler products) which are only available when k is even.

(c) Most importantly, θk(z) is not a cusp form. Arithmetically, we see the inﬂuence of the cusps in the presence of a main term in the asymptotic formula

for Sk(R). Analytically, it implies that the analogues of the inner products

1 hVf , µji and Vf ,E(·, 2 + it) used in the previous chapter no longer converge. Our solution to this problem involves subtracting a linear combination of

Eisenstein series from our analogue of Vf to mollify the behavior at cusps prior to performing a spectral decomposition.

(c0) In a ﬁnal complication, the Eisenstein series one would like to subtract to address (c) in dimension k = 2 are evaluated at poles. In this case, we instead subtract the constant terms of the Laurent series of these Eisenstein series.

We remark that in the process of addressing these technical issues our decompositions for the Dirichlet series

2 2 X Sk(n) X Pk(n) D(s, S × S ) := and D(s, P × P ) := k k ns+k k k ns+k−2 n≥1 n≥1

46 acquire many new terms not present in decompositions of the series D(s, Sf × Sf ). These terms combine and cancel in amazing ways to annihilate would-be poles, cre- ating in the end a dominant singularity in a location consistent with the generalized Gauss Circle Problem on average.

Our work in this chapter builds upon a pair of papers written recently by the author and collaborators Tom Hulse, Chan Ieong Kuan, and David Lowry-Duda; speciﬁcally, this chapter follows [HKLW17a], which addresses the generalized Gauss circle problem in dimension k > 2, and [HKLW17b], which describes the modiﬁcations required by (c0) to complete the case k = 2.

4.1 Algebraic Decompositions

As in the previous chapter (and in particular, as in §3.1), we begin by decomposing the Dirichlet series D(s, Pk × Pk) into a collection of simpler functions. This will be done in two steps: in the ﬁrst, we relate D(s, Pk × Pk) to D(s, Sk × Sk); in the second, we decompose D(s, Sk × Sk) using the same ideas from §3.1.

To simplify notation here and hereafter, we let vk denote the volume of the k-ball

k/2 k of radius one. Explicitly, this means that vk := π /Γ( 2 + 1). Then D(s, Sk × Sk) and D(s, Pk × Pk) are related by the following identity.

47 Proposition 4.1.1. The Dirichlet series D(s, Pk × Pk) and D(s, Sk × Sk) are related through the equality

2 D(s, Pk × Pk) = D(s − 2,Sk × Sk) + vkζ(s − 2)

k k − 2vkζ(s + 2 − 2) − 2vkL(s − 1, θ ) 2v Z Γ(z)Γ(s + k − 2 − z) − k L(s − 1 − z, θk)ζ(z) 2 dz, 2πi k (σ) Γ(s + 2 − 2) where σ > 1 and Re s σ. Here, L(s, θk) denotes the normalized L-function associated to θk(z), deﬁned by

k X rk(n) L(s, θ ) := k . s+ 2 −1 n≥1 n

k/2 Proof. The relation Sk(n) = vkn + Pk(n) implies that

k 2 2 2 2 k Pk(n) = Sk(n) − 2vkn Sk(n) + vkn . (4.1)

Dividing by ns+k−2 and summing over all n ≥ 1 turns the left-hand side of (4.1)

into D(s, Pk × Pk) and turns the ﬁrst term at right in (4.1) into D(s − 2,Sk × Sk).

2 k 2 Similarly, we recognize that the term vkn transforms into vkζ(s − 2).

k/2 As for the sum involving −2vkSk(n)n , we note that

n n−1 X X Sk(n) = rk(m) = 1 + rk(n) + rk(m). m=0 m=1

Summing over n ≥ 1, it follows that

k 2 X Sk(n)n k X rk(n) X X rk(m) s+k−2 = ζ(s + 2 − 2) + k + k n s+ 2 −2 s+ 2 −2 n≥1 n≥1 n n≥1 0

k k X X rk(m) = ζ(s + 2 − 2) + L(s − 1, θ ) + k . (4.2) s+ 2 −2 m≥1 h≥1 (m + h)

48 As in §3.1, we apply the Mellin–Barnes integral transform from Lemma 3.1.1 to decouple the variables m and h from the third term at right in (4.2):

Z k X X rk(m) 1 X X rk(m) Γ(z)Γ(s + − 2 − z) = 2 dz. s+ k −2 s+ k −2−z k 2 2πi 2 z Γ(s + − 2) m≥1 h≥1 (m + h) (σ) m≥1 h≥1 m h 2

For σ > 1, the sum over h ≥ 1 converges absolutely and can be collected into ζ(z); likewise, the sum over m ≥ 1 converges to L(s−1−z, θk) provided that Re s > 2+Re z.

Multiplication by −2vk identiﬁes (4.2) with the second term at right in (4.1) and completes the proof.

Through Proposition 4.1.1 one may pass analytic information from D(s, Sk × Sk)

to D(s, Pk × Pk) (and vice versa) at the cost of understanding a “cross term” which is more or less L(s, θk) and its Mellin–Barnes transform. Since the analytic behavior of L(s, θk) is well known, this cost is negligible.

Our study of D(s, Pk × Pk) thus follows secondhand from the direct study of

D(s, Sk × Sk). This is helpful because D(s, Sk × Sk) is a more natural analogue (at

least, algebraically) of the function D(s, Sf × Sf ) studied in the cusp form case.

As with D(s, Sf × Sf ), we begin our study of D(s, Sk × Sk) by decomposing it into a sum of simpler functions.

2 Proposition 4.1.2. The Dirichlet series associated to Sk(n) decomposes as

D(s, Sk × Sk) = ζ(s + k) + Wk(s) 1 Z Γ(z)Γ(s + k − z) + Wk(s − z)ζ(z) dz 2πi (σ) Γ(s + k)

49 for σ > 1 and Re s > σ + 1, in which

2 X rk(n) W (s) = + 2Z (s + k + 1, 0), (4.3) k ns+k k 2 n≥1

X X rk(n + h)rk(n) Zk(s, w) = k . (4.4) s+ 2 −1 w h≥1 n≥0 (n + h) h

Remark 4.1.3. The statement of Proposition 4.1.2 is essentially identical to that of

Proposition 3.1.2, with Wk(s) in place of Wf (s) and Zk(s, w) in place of Zf (s, w). Indeed, the only diﬀerence between these two results is the presence of the term

ζ(s + k) here and the fact that the n-sum in the deﬁnition of Zk(s, w) includes n = 0.

These two modiﬁcations are necessary because the partial sum deﬁning Sk(n)

k begins with the summand rk(0) and not rk(1). If θ (z) were a cusp form, this diﬀerence

would disappear. Apart from the manual extraction of the terms with rk(0), the proof of Proposition 4.1.2 is identical to that of Proposition 3.1.2, so we omit it here.

4.2 Spectral Expansion of Zk(s, w)

The goal of the next three sections is to develop the meromorphic continuation of

D(s, Sk × Sk). From Proposition 4.1.2, this is reduced to the study of the kernel

Wk(s) deﬁned in (4.3). This function is the sum of a diagonal term,

2 X rk(n) , ns+k n≥1

and the shifted convolution Zk(s, w) defined in (4.4). As in the cusp form case, the diagonal is well understood and the difficulty lies in understanding the shifted convolution. In this section and the next, we produce a meromorphic continuation for Zk(s, w) using a modified version of the technique applied in the cusp form case.

50 4.2.1 Spectral Expansion of the Shifted Convolution Zk(s, w)

We will come to understand the shifted convolution

X X rk(n + h)rk(n) Zk(s, w) = k s+ 2 −1 w h≥1 n≥0 (n + h) h

by ﬁrst ﬁxing a single h and recognizing the sum over n ≥ 0 as a Petersson inner

product of the Poincar´eseries Ph(z, s) against an appropriate modular form; namely,

Z D k 2 k E k 2 k |θ (·)| Im(·) 2 ,Ph(·, s) = |θ (z)| Im(z) 2 Ph(z, s) dµ(z). (4.5) Γ0(4)\H

Note that the Poincar´eseries and the Petersson inner product used in (4.5) are deﬁned

with respect to Γ0(4) instead of SL2(Z), as was the case in Chapter3. A standard unfolding shows that (4.5) may be written

k D k 2 k E Γ(s + 2 − 1) |θ (·)| Im(·) 2 ,Ph(·, s) = Dk(s; h), s+ k −1 (4π) 2

in which Dk(s; h) is deﬁned as the singly-summed shifted convolution

∞ X rk(m + h)rk(m) Dk(s; h) = , s+ k −1 m=0 (m + h) 2

provided that Re s is suﬃciently large. Dividing by hw and summing over h ≥ 1 recovers Zk(s, w):

s+ k −1 k 2 k X Dk(s; h) (4π) 2 X h|θ (·)| Im(·) 2 ,Ph(·, s)i Z (s, w) = = . (4.6) k hw Γ(s + k − 1) hw h≥1 2 h≥1

It is at this point that the direct correspondence with the cusp form case breaks down. While it is tempting to produce a spectral expansion for Zk(s, w) by replacing Ph(·, s) with its spectral decomposition, this is not possible since the function

51 k 2 k 2 |θ (z)| Im(z) 2 is not in L (Γ0(4)\H).

k 2 k We rectify this by subtracting from |θ (z)| Im(z) 2 a linear combination of Eisen-

stein series associated to the cusps of Γ0(4), chosen to cancel the polynomial growth

k 2 k of |θ (z)| Im(z) 2 near each of its cusps.

k 2 k 4.2.2 Modifying |θ (z)| Im(z) 2 to be Square Integrable

For each cusp a of Γ0(4) we deﬁne an associated Eisenstein series, given by

X −1 s Ea(z, s) = Im(σa γz) ,

γ∈Γa\Γ0(4)

in which Γa is the stabilizer of the cusp a and σa ∈ PSL2(R) satisﬁes σa∞ = a and ∼ induces the isomorphism Γa = Γ∞ via conjugation. Following [DI83], these Eisenstein series have Fourier expansions of the form

1 s 1 Γ(s − 2 ) 1−s E (σ z, s) = δ y + π 2 ϕ (s)y a b [a=b] Γ(s) ab0 s 1 (4.7) 2π y 2 X 1 s− 2 + |n| ϕabn(s)Ks− 1 (2π|n|y)e(nx), Γ(s) 2 n6=0

in which the coeﬃcients ϕabn(s) are known but often involve complicated exponential

sums. When b = ∞ (as will often be the case), we write these coeﬃcients as ϕan(s). From (4.7) and asymptotics of the K-Bessel function, we have

k−1 k 1 Γ( ) k k 2 2 2 k 1− 2 −2πy Ea(σbz, 2 ) = δ[a=b]y + π k ϕab0( 2 )y + Ok e (4.8) Γ( 2 )

as Im z → ∞, provided that k 6= 2. The caveat k 6= 2 is required because the function

ϕab0(s) has a singularity at s = 1. This analytic issue, which underlies technical diﬃculty (c0) from page 46, requires special dispensation and will be treated at the

52 end of this chapter in §4.6.

k For now, we assume that k ≥ 3, and remark that (4.8) implies that Ea(σbz, 2 ) vanishes as Im z → ∞ except when a = b, in which case it converges polynomially

k fast to y 2 .

Lemma 4.2.1. For k ≥ 3, the function Vk(z) deﬁned by

k k 2 2 k k Vk(z) := |θ (z)| Im(z) − E∞(z, 2 ) − E0(z, 2 )

2 vanishes at each of the cusps of Γ0(4). Thus Vk(z) ∈ L (Γ0(4)\H).

Proof. The hyperbolic surface Γ0(4)\H has three inequivalent cusps, which we rep-

k 2 k resent by the points ∞, 0, and 1/2. We compute to the growth of |θ (z)| Im(z) 2 at each of these three cusps and compare to that of the Eisenstein series.

The cusp at ∞ is the simplest to address. From the Fourier expansion of θk(z), we obtain

k 2 k k −2πy k k −2πy |θ (z)| Im(z) 2 = y 2 rk(0) + O e = y 2 + O y 2 e

k as y = Im z → ∞. Thus growth at the ∞ cusp is cancelled by subtracting E∞(z, 2 ).

0 − 1 For the cusp at 0, we use the scaling matrix σ = 2 and compute 0 2 0

1 − 1 0 − − 1 1 θ| (z) = (−2iz) 2 θ 2 z = (−2iz) 2 θ − σ0 2 0 4z

− 1 1 = (−2iz) 2 (−2iz) 2 θ(z) = θ(z),

53 in which we’ve applied the involution identity θ(−1/4z) = (−2iz)1/2θ(z) to simplify.

k 2 k k −2πy In particular, |θ (σ0z)| Im(σ0z) 2 = y 2 (1 + O(e )) as Im z → ∞, so subtracting

k E0(z, 2 ) neutralizes growth at the 0 cusp.

1 0 1 Finally, we use the scaling matrix σ 1 = to address the cusp . We have 2 2 1 2

1 1 0 1 z − 2 − 2 θ|σ 1 (z) = (2z + 1) θ z = (2z + 1) θ 2 2 1 2z + 1 1 2 − 1 −1 i 1 1 = (2z + 1) 2 θ = θ − − , −2 − 1/z 2z 2 4z

in which the ﬁnal equality comes from applying the involution equation once more.

1 What remains can be simpliﬁed using the identity θ(z − 2 ) = 2θ(4z) − θ(z), which is easily veriﬁed by comparing the Fourier series of each term. Thus

1 i 2 1 1 z θ|σ 1 (z) = 2θ − − θ − = θ − θ(z), 2 2z z 4z 4

in which the ﬁnal equality again follows from the involution equation for θ(z). Thus

1 −πy/2 θ(σ 1 z) Im(σ 1 z) 4 = O e 2 2

k 1 k 2 2 as Im z → ∞, which implies that |θ (z)| Im(z) → 0 as z → 2 . In particular, it is k 1 k 2 2 not necessary to modify the behavior of |θ (z)| Im(z) at the cusp 2 .

Remark 4.2.2. The behavior of θk(z) at the cusps 0 and 1/2 can also be seen by

k k computing the Fourier expansions at inﬁnity of the functions θ |σ0 (z) and θ |σ1/2 (z).

54 We have

∞ k k X θ |σ0 (z) = θ (z) = rk(n)e(nz), n=0 ∞ k k z X θ |σ 1 (z) = θ − θ(z) = tk(n)e(nz), 2 4 n=0

in which tk(n) is the number of representations of n as the sum of k odd squares. In

k 1 particular, tk(0) = 0 for all k ≥ 1, so θ (z) vanishes at the cusp 2 . This alternative approach reminds us that it is possible to work with θk(z) in a very explicit way.

k 2 k If Vk(z) is used in place of |θ (z)| Im(z) 2 in the inner product against the Poincar´e series in §4.2.1, one obtains

s+ k −1 (4π) 2 k hVk(·),Ph(·, s)i Γ(s + 2 − 1) k k k k (2π) Γ(s − 2 ) ϕ∞h( 2 ) + ϕ0h( 2 ) = Dk(s; h) − . k s− k Γ( 2 )Γ(s) h 2

k Here, we have explicitly computed the inner products hEa(·, 2 ),Ph(·, s)i where they appear by unfolding and applying formula [GR07, §6.621(3)] to evaluate the y-integral of the inner product. Dividing by hw , summing over h ≥ 1, and rearranging yields a formula for Zk(s, w) which is more amenable to spectral expansion than (4.6):

k k k k k s+ 2 −1 (4π) X hVk,Ph(·, s)i (2π) Γ(s − 2 ) X ϕ∞h( 2 ) + ϕ0h( 2 ) Zk(s, w) = + . k w k s+w− k Γ(s + − 1) h Γ( )Γ(s) 2 2 h≥1 2 h≥1 h (4.9)

55 4.2.3 Applying the Spectral Expansion of the Poincar´eSeries

By Selberg’s Spectral Theorem (as in [IK04, Theorem 15.5]), the Poincar´eseries

Ph(z, s) on Γ0(4) has a spectral expansion of the form

X Ph(z, s) = hPh(·, s), µjiµj j (4.10) X 1 Z ∞ + hP (·, s),E (·, 1 + it)iE (z, 1 + it)dt, 4π h a 2 a 2 a −∞

in which a ranges over the cusps of Γ0(4) and {µj} denotes an orthonormal basis

2 1 of Maass forms for L (Γ0(4)\H) with associated types 2 + itj. Note that we have omitted the constant Maass form, since Ph(z, s) is orthogonal to constants. These Maass forms have Fourier expansions,

X 1 2 µj(z) = ρj(n)y Kitj (2π|n|y)e(nx), n6=0 and associated L-functions of the form

X ρj(n) L(s, µ ) = . j ns n≥1

2 Remark 4.2.3. We use the same notation for Maass forms on L (Γ0(4)\H) as was

2 used in Chapter3 for Maass forms on L (SL2(Z)\H). In each case, the level is set beforehand and kept ﬁxed, so we hope that this abuse of notation will not lead to confusion. Note, however, that we cannot always factor out ρj(1) from the L-function in this case due to the existence of oldforms in level 4.

We now produce a spectral expansion for Zk(s, w) by substituting the spectral expansion for the Poincar´eseries into (4.9). Our result is recorded in the following proposition.

56 Proposition 4.2.4. The shifted convolution sum Zk(s, w) has a spectral expansion of the form

k k k k (2π) Γ(s − 2 ) X ϕ∞h( 2 ) + ϕ0h( 2 ) Zk(s, w) = (4.11) k s+w− k Γ( )Γ(s) 2 2 h≥1 h k (4π) 2 X + G(s, it )L(s + w − 1 , µ )hV , µ i (4.12) 2 j 2 j k j j k 1 1 2 Z 2 +z (4π) X G(s, z)π X ϕah( 2 − z) 1 + hV,Ea(·, − z)i dz, (4.13) 1 s+w− 1 −z 2 4πi Γ( + z) 2 a (0) 2 h≥1 h

in which G(s, z) denotes the collected gamma factors,

1 1 Γ(s − 2 + z)Γ(s − 2 − z) G(s, z) := k . Γ(s + 2 − 1)Γ(s)

The three terms at right in (4.11)-(4.13), will be called the “non-spectral part,” the “discrete spectral part,” and the “continuous spectral part,” respectively.

Proposition 4.2.4 should be compared to Proposition 3.2.1, an analogous statement from the cusp form case. The main diﬀerences between these two results are the inclusion of the non-spectral term (4.11), the slight redeﬁnition of G(s, z) to account for the change from weight k to weight k/2, and the less explicit (for now) nature of the continuous spectral part.

Proof. We substitute (4.10) into (4.9) and simplify. This splits the ﬁrst term in (4.9) into two terms (coming from the two terms at right in (4.10)) but does not aﬀect the second term at right in (4.9), which we recognize as (4.11).

57 The contribution of the discrete part of the spectrum of Ph(z, s) simpliﬁes exactly as in the cusp form case. Beginning with the inner product calculation

√ 1 1 ρj(h) π Γ(s − 2 + itj)Γ(s − 2 − itj) hPh(·, s), µji = . s− 1 (4πh) 2 Γ(s) from (3.8), we conclude that these terms contribute

k √ 1 1 s+ 2 −1 (4π) X X ρj(h) π Γ(s − 2 + itj)Γ(s − 2 − itj) hVk, µji. k s+w− 1 Γ(s + − 1) 2 Γ(s) 2 h≥1 j (4πh)

Simpliﬁcation yields the discrete spectral part (4.12).

To address the contribution of the continuous part of the spectrum of Ph(z, s), we must understand the inner products hPh(·, s),Ea(·, w)i for each Eisenstein series on Γ0(4). As in the cusp form case, a standard unfolding (and the Fourier expansion from (4.7)) gives

w+ 1 2π 2 w− 1 Γ(s + w − 1)Γ(s − w) hPh(·, s),Ea(·, w)i = h 2 ϕah(w) , s− 1 (4πh) 2 Γ(s)Γ(w)

1 1 1 so long as Re s > | Re w − 2 | + 2 . Specializing to w = 2 + it with t ∈ R, we obtain

1−it 1 1 1 1 2π ϕah( 2 + it) Γ(s − 2 − it)Γ(s − 2 + it) Ph(·, s),Ea(·, + it) = , 2 s− 1 it 1 (4πh) 2 h Γ(s)Γ( 2 − it)

1 provided that Re s > 2 . Thus the continuous part of the spectrum of Ph(z, s) takes the form

Z ∞ 1 1 1 1 X ϕah( − it)Γ(s − − it)Γ(s − + it) 2 2 2 E (z, 1 + it) dt. 1 1 a 2 2 s− 2 it a −∞ (4πh) (πh) Γ(s)Γ( 2 − it)

Substituting into (4.9) and simplifying yields the continuous spectral term (4.13).

58 4.3 Meromorphic Continuation of Zk(s, w)

In this section, we apply Proposition 4.2.4 to produce a meromorphic continuation for the shifted convolution Zk(s, w). Our analysis follows the decomposition of Zk(s, w) into the non-spectral part (4.11), the discrete spectral part (4.12), and the continuous spectral part (4.13), and we treat each term separately.

4.3.1 Meromorphic Continuation of the Non-Spectral Part

For convenience, we reproduce the non-spectral part of Zk(s, w) below:

k k k k (2π) Γ(s − ) X ϕ∞h( ) + ϕ0h( ) 2 2 2 . k s+w− k Γ( )Γ(s) 2 2 h≥1 h

In its current form, the non-spectral part is not explicit enough for our purposes, and to continue, we must understand the coeﬃcients ϕah(s). While the non-spectral term only requires ϕ∞h(s) and ϕ0h(s), we will treat all three cusps at the same time. (The

1 cusp 2 will be required when we address the continuous spectral term (4.13), anyway.)

When b = ∞ and the cusp a is represented in the form a = u/v with (u.v) = 1, the exact deﬁnition of the coeﬃcients ϕabh(t) is given by [DI83] as

t ∞ (v, 4/v) X X hδ ϕ (t) = γ−2t e . (4.14) ah 4v γv (γ,4/v)=1 δ(γv)∗ γδ≡u mod (v,4/v)

In general, this is a Dirichlet series whose coeﬃcients are modiﬁed Ramanujan sums.

1 The three inequivalent cusps of Γ0(4), which we have referred to as 0, 2 , and ∞, 1 1 can be represented by 1, 2 , and 4 , respectively. It is a standard exercise (by adapting

59 the technique of [Gol15, §3.1], e.g.) to compute these coeﬃcients, and we ﬁnd that

(2) h (2) σ1−2t(h) (−1) σ1−2t(h) ϕ0h(t) = , ϕ 1 h(t) = , 4tζ(2)(2t) 2 4tζ(2)(2t) (4.15) 22−4tσ ( h ) − 21−4tσ ( h ) ϕ (t) = 1−2t 4 1−2t 2 , ∞h ζ(2)(2t)

(2) in which σν(h) is the sum of divisors function, σν (h) is the sum of odd-divisors function, and ζ(2)(t) is the Riemann zeta function with the 2-factor of its Euler product removed. To give an example of one of these calculations, we present the computation

of ϕ0h(t) in the following lemma.

Lemma 4.3.1. For Re t 1, we have

σ(2) (h) ϕ (t) = 1−2t . 0h 4tζ(2)(2t)

Proof. From (4.14), we obtain

∞ 1 X cγ(h) X uh ϕ (t) = , in which c (h) := e 0h 4t γ2t q q (γ,2)=1 u(q)∗

denotes the Ramanujan sum. We can sift out the integers u relatively prime to q using the M¨obiusfunction; speciﬁcally,

q q q/d X uh X X X X uh X X hmd c (h) = e µ(d) = µ(d) e = µ(d) e . q q q q u=1 d|q,d|u d|q u=1 u≡0(d) d|q m=1

This last sum is 0 unless (q/d) | h, in which case it evaluates to q/d. Thus

X X q X q c (h) = µ(d) µ(d) = `µ , q d ` d|q q `|h,`|q d |h

60 by change of variable ` = q/d. Finally, we compute

∞ ∞ 1 X X γ 1 X X µ(m) ϕ (t) = γ−2t `µ = ` 0h 4t ` 4t (m`)2t (γ,2)=1 `|h,`|γ `|h (m,2)=1 (`,2)=1 (2) 1 X σ (h) = `1−2t · ζ(2)(2t)−1 = 1−2t , 4t 4tζ(2)(2t) `|h (`,2)=1

as claimed.

Remark 4.3.2. The computations for the cusps ∞ and 1/2 are similar. Each case, including the one presented above, can be thought of as an extension of the identity

∞ X cq(h) σ1−s(h) = qs ζ(s) q=1

from [Gol15, Proposition 3.1.7], with a suitable modiﬁcation in the 2-factors.

Returning to the non-spectral part, we note that by dividing by hw and summing over h ≥ 1, we have

(2) X ϕ0h(t) ζ(w)ζ (w − 1 + 2t) = , hw 4tζ(2)(2t) h≥1 1−w (2) X ϕ 1 h(t) (2 − 1)ζ(w)ζ (w − 1 + 2t) 2 = , hw 4tζ(2)(2t) h≥1 X ϕ∞h(t) ζ(w)ζ(w − 1 + 2t) 1 1 = − . hw 24tζ(2)(2t) 4w−1 2w−1 h≥1

By substitution, we see that the non-spectral part of Zk(s, w) has the explicit form

πkΓ(s − k )ζ(s + w − k )ζ(s + w + k − 1) 4 4 2 2 2 1 + − . k (2) 2s+2w k 2 2 +s+w Γ( 2 )Γ(s)ζ (k) 2

k k This term is analytic in the region Re s > 2 and Re(s + w) > 1 + 2 and extends 2 k meromorphically to all of C . The non-spectral part has polar lines at s + w = 1 + 2 61 k k and s + w = 2 − 2 as well as isolated poles in s at the poles of Γ(s − 2 )/Γ(s).

k In the distinguished case w = 0, we note analyticity in the half-plane Re s > 1 + 2 k and the potential for poles at s = 1 + 2 − j for each integer j ≥ 0.

4.3.2 Meromorphic Continuation of the Discrete Spectral

Part

The discrete spectral part of Zk(s, w), which takes the form

k (4π) 2 X G(s, it )L(s + w − 1 , µ )hV , µ i, 2 j 2 j k j j

2 1 is clearly meromorphic in C and analytic in s, w in the region Re s > 2 + sup | Im tj|, in which the supremum runs over all types tj on Γ0(4). Partial progress towards

Selberg’s Eigenvalue Conjecture due to [Hux85] gives sup | Im tj| = 0, hence the

1 discrete spectral part of Zk(s, w) is analytic in Re s > 2 .

1 Moreover, Zk(s, 0) is analytic in the region Re s > − 2 . This follows from the two lemmas presented in §3.3.1, which generalize to this setting with minimal changes. That is,

a. For odd Maass forms, hVk, µji = 0. (This follows from Lemma 3.3.2 and the fact that Maass forms are orthogonal to Eisenstein series.)

b. For even Maass forms, L(−2m ± itj, µj) = 0 for all m ≥ 0.

Specializing the discrete spectral part to w = 0, these facts imply that the apparent

1 poles on the line Re s = 2 do not actually occur. We conclude that the discrete 1 spectral part of Zk(s, 0) is meromorphic in C and analytic in the half-plane Re s > − 2 .

62 4.3.3 Meromorphic Continuation of the Continuous Spectral

Part

As in the cusp form case, the meromorphic continuation of the continuous spectral is

the most delicate. Fortunately, the argument for addressing Zk(s, w) is quite similar to that used in §3.3.2 for the cusp form case, so we can safely omit some laborious details and focus on introducing notation and presenting results.

For simplicity, we write the continuous spectral part (4.13) in the form

k 1 +z (4π) 2 X Z G(s, z)π 2 ζ (s + w, z) V ,E (·, 1 − z) dz, 4πi Γ( 1 + z) a k a 2 a (0) 2

in which ζa(s, z) is deﬁned by

1 X ϕah( 2 − z) ζa(s, z) = 1 . (4.16) s− 2 −z h≥1 h

These functions are rough analogues of the “collected zeta factors” ζ(s, w, z) from the cusp form case and are deﬁned, explicitly, by the equations

ζ(s − 1 − z)ζ(2)(s − 1 + z) ζ (s, z) = 2 2 , 0 21+2zζ(2)(1 + 2z) 1 (2) 1 z ζ(s + 2 − z)ζ (s − 2 + z) 2 ζ 1 (s, z) = 3 − 1 , (4.17) 2 1+2z (2) s− 2 ζ (1 + 2z) 2 2 1 1 z z ζ(s − 2 − z)ζ(s − 2 + z) 4 2 ζ∞(s, z) = − . 2+4z (2) s− 3 s− 3 2 ζ (1 + 2z) 4 2 2 2

By inspection, we see that the continuous spectral part of Zk(s, w) is analytic in the

3 1 intersection of the half-spaces Re(s + w) > 2 and Re s > 2 .

To produce a meromorphic continuation of the continuous spectral part beyond this region, we argue as in §3.3.2, shifting and unshifting lines of integration and 63 extracting pairs of residual terms. Fortunately, the analogy is quite exact: for each

cusp a, the function ζa(s + w, z) has potential poles and polar lines at the same locations as that of ζ(s, w, z), and the rest of the continuous spectral term is identical. (The sum over the cusps does nothing more than complicate simpliﬁcation later on.)

We conclude that the continuous spectral part of Zk(s, w) has a meromorphic con-

tinuation to all of C2, obtained by accounting for appropriate residual pairs. Adapting the proof of Proposition 3.3.4 produces the following.

Proposition 4.3.3. The continuous spectral part of Zk(s, w) has a meromorphic

continuation to all of C2. This continuation may be written explicitly as

k 1 +z (4π) 2 X Z G(s, z)π 2 ζ (s + w, z) V ,E (·, 1 − z) dz 4πi Γ( 1 + z) a k a 2 a (0) 2

+ − X + − + δ[Re(s+w)< 3 ] T 3 (s, w) − T 3 (s, w) + T 1 (s, w) − T 1 (s, w) , 2 2 2 2 −j 2 −j 1 j< 2 −Re s

in which

k 1 2 2 +z + (4π) X G(s, z)π 1 T 3 (s, w) = Res 1 ζa(s + w, z)hVk,Ea(·, 2 − z)i, 2 2 z= 3 −s−w Γ( + z) a 2 2 k 1 2 2 +z − (4π) X G(s, z)π 1 T 3 (s, w) = Res 1 ζa(s + w, z)hVk,Ea(·, 2 − z)i, 2 2 z=s+w− 3 Γ( + z) a 2 2

and, for each j ≥ 0, we deﬁne

k 1 +z (4π) 2 X G(s, z)π 2 T + (s, w) = Res ζ (s + w, z)hV ,E (·, 1 − z)i, 1 −j 1 a k a 2 2 2 z= 1 −j−s Γ( + z) a 2 2 k 1 +z (4π) 2 X G(s, z)π 2 T − (s, w) = Res ζ (s + w, z)hV ,E (·, 1 − z)i. 1 −j 1 a k a 2 2 2 z=s+j− 1 Γ( + z) a 2 2

We recall from the cusp form case that the pair of residues R+(s, w) and R−(s, w) are additive opposites. This is likewise true for T ±(s, w), at least in the case w = 0, 64 but is harder to verify because each residue is deﬁned as a sum over the three cusps

a of Γ0(4) and because the functional equations for Eisenstein series on Γ0(4) are more complicated.

To describe the functional equations of the family of Eisenstein series on Γ0(4), let E(z, s) = {Ea(z, s)}a. Following Iwaniec [Iwa02, §13.3], we have

E(z, s) = Φ(s)E(z, 1 − s),

in which Φ(s) denotes the symmetric scattering matrix

√ πΓ(s − 1 ) Φ(s) = 2 ϕ (s) Γ(s) ab0 a,b

composed of (essentially) the constant Fourier coeﬃcients of the family of Eisenstein

series Ea(σbz, s). Explicitly, this implies that

      (2) (2) E (z, s) ζ(2s−1) ζ (2s−1) ζ (2s−1) E (z, 1 − s)  ∞  √  24s−1 4s 4s   ∞    1       π Γ(s − 2 )  (2) (2)     E (z, s)  =  ζ (2s−1) ζ(2s−1) ζ (2s−1)   E (z, 1 − s)  .  0  Γ(s)ζ(2)(2s)  4s 24s−1 4s   0           ζ(2)(2s−1) ζ(2)(2s−1) ζ(2s−1)    E 1 (z, s) s s 4s−1 E 1 (z, 1 − s) 2 4 4 2 2

We can use the functional equation of the Eisenstein series to show that the residue

± ± pairs T (s, 0) are related. We separate the case T 3 and treat it ﬁrst. 2

+ − Lemma 4.3.4. We have T 3 (s, 0) = −T 3 (s, 0). 2 2

− Proof. The residue T3/2(s, 0) is given as a sum of three terms, indexed by the three

cusps a of Γ0(4). Fortunately, two of these terms – those corresponding to the cusps at a = ∞ and a = 1/2 – vanish due to the nature of the 2-factors in their collected

65 zeta functions ζa(s, z). Focusing on the a = 0 term and simplifying, we conclude that

k s+w−1 − (4π) 2 Γ(1 − w)Γ(2s + 2w − 2)π hVk,E0(·, 2 − s − w)i T 3 (s, w) = − k . (4.18) 2 2s+w−1 2 Γ(s)Γ(s + 2 − 1)Γ(s + w − 1)

Specializing to the case w = 0 and applying the Gauss duplication formula gives

k 1 s− 3 2 2 − (4π) Γ(s − 2 )π hVk,E0(·, 2 − s)i T 3 (s, 0) = − · k . 2 2 2Γ(s)Γ(s + 2 − 1)

Applying the Gauss duplication formula and the functional equations of E0(z, s) and

− the Riemann zeta function transforms T3/2(s, 0) into

k 2−s (4π) 2 Γ(2s − 2)π ζ(2s − 2) hV ,E (·, s − 1)i − × k 0 k (2) 25−2s 2Γ(s)Γ(s + 2 − 1)Γ(2 − s)ζ (4 − 2s) 3−2s (43−2s − 23−2s)hV ,E (·, s − 1)i (2 − 1)hVk,E 1 (·, s − 1)i + k ∞ + 2 . 28−4s 25−2s

+ Simpliﬁcation shows that this is equal to −T3/2(s, 0), which completes the lemma.

Further residual pairs may be treated in a uniform manner.

+ − Lemma 4.3.5. We have T 1 (s, 0) = −T 1 (s, 0). 2 −j 2 −j

Proof. The proof of this lemma is similar to that of Lemma 4.3.4, in that it follows from the functional equation of the Eisenstein series. On the other hand, it is more computational, since each of the three cusps contribute.

± We will not require explicit formulas for any of the residual pairs T1/2−j for j > 0. − In the case j = 0, we obtain a particularly simple formula for T1/2(s, 0) by observing that the two summands corresponding to the cusps a = 0 and a = 1/2 vanish. This is

1 1 easily seen through (4.17), which shows that ζ0(s, s− ) and ζ 1 (s, s− ) are multiplied 2 2 2

66 by ζ(2)(0) = 0. Simpliﬁcation gives

k 1−s + (4π) 2 π Γ(2s) ζ(2s − 1) T 1 (s, 0) = k · hVk,E∞(·, s)i. 2 ζ(2 − 2s) 4Γ(s)Γ(s + 2 − 1)Γ(1 − s)

Applying the functional equation of the Riemann zeta function and the Gauss duplication formula, this further simpliﬁes to

s+ k −1 + (4π) 2 T 1 (s, 0) = − k hVk,E∞(·, s)i. (4.19) 2 2Γ(s + 2 − 1)

4.4 Analytic Behavior of Wk(s)

In this section, we apply the meromorphic continuation of Zk(s, 0) given in Proposi- tion 4.3.3 to study the analytic properties of the kernel function Wk(s). Since Wk(s) is the sum of a diagonal term and a shifted convolution,

2 X rk(n) W (s) = + 2Z (s + k + 1, 0), k ns+k k 2 n≥1 we begin by discussing the analytic properties of the diagonal part. We then demon-

k strate cancellation between Zk(s + 2 + 1, 0) and the diagonal and conclude with

Theorem 4.4.2, which describes the rightmost poles of Wk(s).

4.4.1 The Diagonal Part

The analytic properties of the diagonal part are obtained by recognizing it in terms of the Rankin–Selberg L-function associated to θk × θk, deﬁned by

2 k k X rk(n) L(s, θ × θ ) = ζ(2s) k . s+ 2 −1 n≥1 n

67 k k 2 k Unfortunately, y 2 |θ (z)| is not of rapid decay, since θ (z) is not a cusp form, which means that the classical inner product formula for the Rankin–Selberg convolution (as described in §2.3) does not converge. This technical issue was shored up by

Zagier’s regularization method [Zag81] in the case of modular forms on SL2(Z) and extended to modular forms on congruence subgroups by Gupta [DG00].

Following Gupta, we have the identiﬁcation

k k k Z ∞ Γ(s + 2 − 1) L(s, θ × θ ) s−1 dy = (c0(y) − ψ0(y)) y (4.20) s+ k −1 (4π) 2 ζ(2s) 0 y

in some vertical strip (to be speciﬁed), in which c0(y) is the constant Fourier coeﬃcient of Vk(σ0z) and ψ0(y) is a linear combination of products of powers of y and log y for

−N which Vk(σ0z) satisﬁes Vk(σ0z) = ψ0(y)+O(y ) for all N > 0 in the limit as y → ∞.

1− k In our particular instance, ψ0(y) consists only of the constant multiple of y 2

k which appears in the constant coeﬃcient of the Eisenstein series E0(z, 2 ). Following

Gupta, we may therefore identify the left-hand side of (4.20) with hVk(σ0z),E0(·, s)i

k k for s in the vertical strip 1 − 2 < Re s < 2 . Moreover, (4.20) has a meromorphic continuation to C, a functional equation of the form s 7→ 1 − s, and admits potential k k poles only at s = 2 , 1, 0, 1 − 2 , and the zeros of ζ(2s).

We conclude that the diagonal part of Wk(s) has potential poles at s =

k k −1, − 2 , − 2 − 1, −k, and at the zeros of ζ(2s + k + 2). (This last collection of zeros k+1 k 3 is conﬁned to Re s < − 2 and and will not appear until Re s = − 2 − 4 under the Riemann hypothesis.)

68 The rightmost pole of the diagonal part (at s = −1) appears on the edge of the strip of validity for the convenient representations above. For this reason, we ignore that closed form when calculating the associated residue and instead apply the Wiener–Ikehara Theorem (Theorem 3.5.1) to the series, which converges for Re s > −1 and has s = −1 as its unique pole on the line Re s = −1. We have

∞ 2 k X rk(m) k − 1 X 2 π ζ(k − 1) Res = lim rk(m) = . (4.21) s=−1 ms+k X→∞ Xk−1 ζ(2)(k)Γ( k )2 m=1 m≤X 2

This second equality is the subject of [CKO05], which applies a general method for evaluating sums of positive definite quadratic forms that originates with Müller[Mül92].

k The second rightmost pole occurs at s = − 2 and is best understood through (4.20) and the identiﬁcation of these expressions with hVk(σ0z),E0(·, s)i. We calculate the residue at this point to be

k (4π) 2 Res hVk,E0(·, s)i . k s=1 Γ( 2 )

Further poles may be analyzed using the functional equation for L(s, θk × θk). This information will not be necessary, however, as we will show that the diagonal part

± cancels with the residue pair T1/2(s, 0) in a region which contains these poles.

Remark 4.4.1. The precocious reader may note that the L-function L(s, θk ×θk) has

k a second representation in terms of the inner product hVk,E∞(·, 2 )i. Our preference

for E0 in the formulas above was chosen so as to simplify exposition in future sections.

69 4.4.2 The Discrete Part

k Accounting for the shifted argument in Zk(s + 2 + 1, 0), we note that the discrete k+3 part of Wk(s) is analytic in the right half-plane Re s > − 2 . Arithmetically, we have

little to say that takes advantage of the properties of Wk(s) past a cluster of poles on

k+3 the line Re s = − 2 , so we restrict our analysis to the right of this line.

4.4.3 The Continuous Part

The meromorphic continuation of the continuous part of Wk(s) must account for many pairs of residual terms that appear as Re s decreases. However, only the ﬁrst

± ± k+3 two pairs, T3/2 and T1/2, make an appearance in the right half-plane Re s > − 2 . The integral expression in the continuous part does not contribute poles directly, so we focus on the contribution of these two residual pairs.

The contribution from the rightmost pair takes the form

k 1 s+ k−1 k k Γ(s + + )π 2 hVk,E0(·, 1 − − s)i + k 2 2 2 2 4T 3 (s + 2 + 1, 0) = (4π) k . (4.22) 2 Γ(s + 2 + 1)Γ(s + k)

This term has inﬁnitely many potential poles. Yet of these, no more than two appear

k+3 k in the half-plane Re s > − 2 . The dominant pole occurs at s = − 2 from within the Eisenstein series, with residue

k 2 + k (4π) Res 4T 3 (s + 2 + 1, 0) = − k Res hVk,E0(·, s)i . s=− k 2 s=1 2 Γ( 2 )

A second pole appears at s = 1 − k from the regularization of the inner product, which lies in the relevant region only in the cases k ≤ 4. In the case k = 4, this pole

k −1 is cancelled by a zero in Γ(s + 2 + 1) and does not appear. In the sole remaining case k = 3 (since our method does not support k = 2 without modiﬁcation), this pole

70 k 1 k 1 actually collides with another pole at s = − 2 − 2 coming from Γ(s + 2 + 2 ). Thus we obtain a double pole with principal part

π2 24a ζ(2)(3) − π2γ − π2 log(4π) − + 0 , 3ζ(2)(3)(s + 2)2 3ζ(2)(3) · (s + 2)

in which γ is the Euler–Mascheroni constant and a0 is the constant coeﬃcient of the

3 Laurent expansion of the inner product hV3,E0(·, s)i about the point s = 2 .

k 1 For k ≥ 4, the previously discussed pole in Γ(s + 2 + 2 ) remains in the relevant half-plane but is simple. This pole contributes a residue

3 + k k hVk,E0(·, 2 )i Res 4T (s + + 1, 0) = (4π) 2 . 3 2 3/2 k−1 s=− k+1 2 2 π Γ( 2 )

The second residual pair is easier to discuss. Following (4.19), we note that

s+k k k + k (4π) k L(s, θ × θ ) 2T 1 (s + 2 + 1, 0) = − hVk,E∞(·, s + 2 + 1)i = − 2 Γ(s + k) ζ(2s)

and see that the second residual pair exactly cancels with the diagonal part. We

± note that this cancellation only occurs in the region where T1/2 is included in the k 1 meromorphic continuation of the continuous part, ie. for Re s < − 2 − 2 . In practice, ± this means that we will ignore both the diagonal term and T1/2 once the latter appears.

4.4.4 The Non-Spectral Part

We conclude by considering the non-spectral part. This is quite simple, as this term

k is by now completely explicit. As it appears within 2Zk(s+ 2 +1, 0), the non-spectral part takes the form

2πkΓ(s + 1)ζ(s + 1)ζ(s + k) 1 1 E (s) := 1 + − . (4.23) k k k (2) 22s+k 2s+k−1 Γ( 2 )Γ(s + 2 + 1)ζ (k) 71 Thus Ek(s) is analytic in the region Re s > 0 and extends meromorphically to a function on the entire complex plane. For any choice of k, we see the presence of simple poles at s = 0 and s = −1. Potential poles at negative odd integers (from Γ(s + 1)) are cancelled by trivial zeta zeros, and the existence of poles at negative even integers depends on the choice of k.

When k is odd, Ek(s) admits poles at each negative even integer and a double pole at s = 1 − k which comes from Γ(s + 1)ζ(s + k). When k is even, zeros from

k k ζ(s + 1)ζ(s + k)/Γ(s + 2 + 1) cancel all but b 4 c of the lesser poles, leaving only poles k at 0, −1, and each negative even integer greater than −1 − 2 . The two rightmost poles have residues

2πk πkζ(k − 1) Res Ek(s) = , Res Ek(s) = − . s=0 k k s=−1 (2) k 2 Γ( 2 )Γ( 2 + 1) ζ (k)Γ( 2 )

We note that the residue at s = −1 above and the residue at s = −1 in a pole in the diagonal part from (4.21) sum to 0. These poles therefore cancel.

4.4.5 Polar Analysis of Wk(s)

k+3 At last, we present a polar analysis of Wk(s) in the region Re s > − 2 , the largest right half-plane in which it has ﬁnitely many poles. By collecting our observations on the diagonal, discrete, continuous, and non-spectral parts from §4.4.1-4.4.4, we produce the following theorem.

Theorem 4.4.2. The function Wk(s) has meromorphic continuation to the entire

k+3 complex plane. In the half-plane Re s > − 2 , all but one of the contributing poles of

Wk(s) occur at non-positive even integers and can be read from the non-spectral part

2πkΓ(s + 1)ζ(s + 1)ζ(s + k) 1 1 E (s) := 1 + − . k k k (2) 22s+k 2s+k−1 Γ( 2 )Γ(s + 2 + 1)ζ (k) 72 k+1 The only other polar contribution in this half-plane occurs at s = − 2 and can be + k read from the residual 4T3/2(s + 2 + 1, 0). When k > 3, this pole is simple and has residue 3 + k k hVk,E0(·, 2 )i Res 4T (s + + 1, 0) = (4π) 2 . 3 2 3/2 k−1 s=− k+1 2 2 π Γ( 2 )

+ k For k = 3, this pole is a double pole, and the Laurent series of 4T3/2(s + 2 + 1, 0) about s = −2 has principal part

π2 24a ζ(2)(3) − π2γ − π2 log(4π) − + 0 , 3ζ(2)(3)(s + 2)2 3ζ(2)(3)(s + 2)

in which a0 is the constant term in the Laurent series for the meromorphic continu-

3 ation of hVk,E0(·, s)i centered at s = 2 .

Remark 4.4.3. Since much is known about the generalized Gauss circle problem in higher dimensions (especially, in dimension k ≥ 5), it is particularly interesting to illustrate Theorem 4.4.2 in low dimension. In dimension k = 3, we note that

16π2 π2 24a ζ(2)(3) + π2(γ − log(2π) + 2 − 24ζ0(−1)) W (s) − − − 0 3 3s 3ζ(2)(3)(s + 2)2 3ζ(2)(3)(s + 2) is analytic in Re s > −3. In dimension k = 4, we see that

4 2 3 π 4π 32hV4,E0( 2 )i W4(s) − − − 5 s (s + 2) (s + 2 )

7 is analytic in Re s > − 2 . The inner product in the line above can (and will) be made more explicit using a known factorization for L(s, θ4 × θ4) in terms of classical Dirichlet L-functions. We postpone such discussion until Remark 4.5.1.

73 4.5 Polar Analysis of D(s, Sk × Sk) and D(s, Pk × Pk)

In this section, we compute the locations and singular parts of the rightmost poles of

D(s, Sk ×Sk) and D(s, Pk ×Pk). This will be done in two subsections: in the ﬁrst, we

study D(s, Sk ×Sk) using the properties of Wk(s) described in the previous section; in

the second, we study D(s, Pk ×Pk) by relating it to D(s, Sk ×Sk) via Proposition 4.1.1.

4.5.1 Polar Analysis of D(s, Sk × Sk)

We recall from Proposition 4.1.2 that

1 Z Γ(z)Γ(s + k − z) D(s, Sk × Sk) = ζ(s + k) + Wk(s) + Wk(s − z)ζ(z) dz, 2πi (σ) Γ(s + k)

in which σ > 1 and Re s > σ to begin. Note that the integral is analytic in the region Re s > σ. By shifting (σ) to (−3 + ), we extract three residues from the integral and conclude that

W (s − 1) D(s, S × S ) = ζ(s + k) + W (s) + k − 1 W (s) + 1 (s + k)W (s + 1) k k k s + k − 1 2 k 12 k 1 Z Γ(z)Γ(s + k − z) + Wk(s − z)ζ(z) dz. 2πi (−3+) Γ(s + k)

Here, a potential residue from the pole at z = −2 in Γ(z) is cancelled by a trivial zeta zero. Since the shifted integral is analytic in Re s > −3 + , we may read oﬀ the poles and residues of D(s, Sk ×Sk) in this region by considering the other terms.

In practice, one could conduct a polar analysis of D(s, Sk × Sk) over the entire complex plane. The result would be rather cumbersome and it is not clear that being fully explicit helps in our applications. For this reason, we restrict our analysis to the largest right half-plane for which D(s, Sk × Sk) (and later, D(s, Pk × Pk)), has 74 ﬁnitely many poles.

k+1 For D(s, Sk × Sk), this amounts to Re s > − 2 . For ease of exposition, we make a further concession and present our results in the uniform region Re s > −2. Note

k+1 that this coincides with Re s > − 2 in the interesting case k = 3.

The poles and residues of D(s, Sk × Sk) in the right half-plane Re s > −2 are collected in Table 4.1 for easy reference. For simple poles, these residue computations are immediate in light in Theorem 4.4.2. To compute the singular parts of the double

pole contributions in Wk(s − 1)/(s + k − 1) in the case k = 3, we compute as well the singular part of (s + k − 1)−1 about s = −1 and multiply Laurent polynomials. Thus

E (s − 1) 2π2 π2(2γ + log 2 − 24ζ0(−1)) 3 = + + O(1) (4.24) s + 2 3ζ(2)(3)(s + 1)2 3ζ(2)(3)(s + 1) + 3 4T 3 (s + 2 , 0) π2 π2(1 − γ − log(4π)) 8a 2 = − + + 0 + O(1) (4.25) s + 2 3ζ(2)(3)(s + 1)2 3ζ(2)(3)(s + 1) (s + 1)

as s → −1.

Table 4.1: Summary of Polar Data for D(s, Sk × Sk) in the Half-Plane Re s > −2

pole location source of pole residue

Ek(s−1) Wk(s−1) 2 s = 1 s+k−1 , from s+k−1 vk

1 1 k 2 s = 0 2 Ek(s), from 2 Wk(s) 2 vk

(s+k) 1 1 2 s = −1 12 Ek(s + 1), from 12 (s + k)Wk(s + 1) 12 vkk(k − 1) k 3−k Ek(s−1) Wk(s−1) π ζ(k−2)(1+2 ) s = −1, if k 6= 3 s+k−1 , from s+k−1 k 2 (2) 12Γ( 2 ) ζ (k) Ek(s−1) Wk(s−1) s = 2 − k, k = 3 s+k−1 , from s+k−1 double pole, see (4.24) + k k 4T (s+ ,0) 2 3 k+1 3/2 2 Wk(s−1) (4π) hV,E0(·, 2 )i s = 1 − 2 , k 6= 3 s+k−1 , from s+k−1 3/2 k+1 π Γ( 2 ) 4T + (s+ k ,0) k+1 3/2 2 Wk(s−1) s = 1 − 2 , k = 3 s+k−1 , from s+k−1 double pole, see (4.25)

75 k+1 Remark 4.5.1. The travelling simple pole at s = 1 − 2 described in Table 4.1 for k 6= 3 is particularly interesting in the case k = 4, when it appears in the right half-plane Re s > −2. In this case, Borwein and Choi [BC03] give the explicit analytic continuation

∞ 2 6−3s 3−2s 1−s 2 X r4(m) (2 − 5 · 2 + 2 + 1)ζ(s − 2)ζ(s − 1) ζ(s) = 64 , ms (1 + 21−s)ζ(2s − 2) m=1

3 which can be used to give a closed form representation of hV4,E0(·, 2 )i through Zagier 3 regularization (as in §4.4.1). We conclude that the residue of D(s, S4 × S4) at s = − 2

is given by √ 16(9 2 − 8)ζ( 1 )ζ( 3 )2ζ( 5 ) C0 := 2 2 2 ≈ −12.17821 06119. 4 7π2ζ(3)

4.5.2 Polar Analysis of D(s, Pk × Pk)

We recall from Proposition 4.1.1 that

D(s, Pk × Pk) = D(s − 2,Sk × Sk) (4.26)

2 k k + vkζ(s − 2) − 2vkζ(s + 2 − 2) − 2vkL(s − 1, θ ) (4.27) 2v Z Γ(z)Γ(s + k − 2 − z) − k L(s − 1 − z, θk)ζ(z) 2 dz, (4.28) 2πi k (σ) Γ(s + 2 − 2)

provided that σ > 1 and Re s 1. Thus the meromorphic properties of D(s, Sk ×Sk) determine those of D(s, Pk × Pk), once one accounts for the variety of “correction terms” in lines (4.27) and (4.28).

Since the poles of D(s, Sk × Sk) have been described in full only within the right half-plane Re s > −2 (cf. Table 4.1), we restrict our analysis of D(s, Pk × Pk) to the right half-plane Re s > 0. For ease of reference, our classiﬁcation of the poles of

76 Table 4.2: Summary of Polar Data for D(s, Pk × Pk) in the Half-Plane Re s > 0

pole location line source of pole residue

2 2 s = 3 (4.27) vkζ(s − 2) vk

Ek(s−3) Wk(s−3) 2 s = 3 (4.26) s+k−3 , from s+k−3 vk L(s−2,θk) 2 s = 3 (4.28) −2vk k −2vk s+ 2 −3

1 1 k 2 s = 2 (4.26) 2 Ek(s − 2), from 2 Wk(s − 2) 2 vk k 2 s = 2 (4.27) −2vkL(s − 1, θ ) −kvk k k 2 s = 2 (4.28) vkL(s − 1, θ ) 2 vk

k k s = 3 − 2 (4.27) −2vkζ(s + 2 − 2) −2vk k L(s−2,θk) k k s = 3 − 2 (4.28) −2vk k −2vkL(1 − 2 , θ ) s+ 2 −3

1 1 2 s = 1 (4.26) 12 (s + k − 2)Ek(s − 1) 12 vkk(k − 1) k 3−k Ek(s−3) Wk(s−3) π ζ(k−2)(1+2 ) s = 1, if k 6= 3 (4.26) s+k−3 , from s+k−3 k 2 (2) 12Γ( 2 ) ζ (k) Ek(s−3) Wk(s−3) † s = 4 − k, k = 3 (4.26) s+k−3 , from s+k−3 double pole, see (4.24) + k k 4T (s+ −2,0) 2 3 k+1 3/2 2 Wk(s−3) (4π) hV,E0(·, 2 )i s = 3 − 2 , k 6= 3 (4.26) s+k−3 , from s+k−3 3/2 k+1 π Γ( 2 ) 4T + (s+ k −2,0) k+1 3/2 2 Wk(s−3) † s = 3 − 2 , k = 3 (4.26) s+k−3 , from s+k−3 double pole, see (4.25) k k k/2 k L(s,θ )(s+ 2 −2) π ( 2 −1) s = 1 (4.28) −2vk 12 −vk 6Γ(k/2) † The Laurent series in (4.24) and (4.25) must be recentered to s = 1 via s 7→ s − 2.

D(s, Pk × Pk) is summarized in Table 4.2.

Polar terms which arise from (4.26) can be read oﬀ from Table 4.1 and the meromorphic behavior of the functions in (4.27) is well known. To verify our claims about the poles of (4.28), we shift the line of integration in (4.28) to (−3+). By the residue

77 theorem, (4.28) equals

v Z Γ(z)Γ(s + k − 2 − z) − k L(s − 1 − z, θk)ζ(z) 2 dz (4.29) πi k (−3+) Γ(s + 2 − 2) ! L(s − 2, θk) L(s − 1, θk) L(s, θk)(s + k − 2) − 2v − + 2 . (4.30) k k 2 12 s + 2 − 3

The shifted integral (4.29) is analytic in Re s > −1+, so the relevant polar terms can be read from the residues in (4.30). We detect three poles which arise from the

k k pole at s = 1 in L(s, θ ) and an additional pole at s = 3 − 2 which comes from the k k −1 denominator of L(s − 2, θ )(s + 2 − 3) .

4.5.3 Cancellation in the Poles of D(s, Pk × Pk)

We now demonstrate cancellation in the poles of D(s, Pk × Pk). Our results are presented in Theorem 4.5.2, which classiﬁes the meromorphic behavior of D(s, Pk × Pk) in the right half-plane Re s > 0.

With reference to Table 4.2, we note that the potential poles at s = 3 and s = 2

k cancel. Likewise, the potential poles at s = 3 − 2 cancel. To see this last fact, we k k k evaluate L(1 − 2 , θ ) using the functional equation of L(s, θ ),

k −s− 2 +1 k k s−1 k k π Γ(s + 2 − 1)L(s, θ ) = π Γ(1 − s)L(2 − 2 − s, θ ), (4.31) and conclude that

Γ( k ) L(1 − s, θk) Γ( k ) L(1 − k , θk) = 2 lim = − 2 Res L(s, θk) = −1. (4.32) 2 πk/2 s→0 Γ(s) πk/2 s=1

k It is now clear from Table 4.2 that the potential pole at s = 3 − 2 does not appear.

78 The remaining polar terms in Table 4.2 do not cancel, and as such represent the

dominant singularities of D(s, Pk ×Pk). By combining these terms when they appear, we produce the following theorem.

Theorem 4.5.2. The Dirichlet series D(s, Pk × Pk), originally deﬁned as a series in

the right half-plane Re s > 3, has a meromorphic continuation to C given by (4.26)- (4.28) and is analytic in the right half-plane Re s > 1, with a pole at s = 1. In dimension k ≥ 4 this pole is simple, with residue

k2v2 πkζ(k − 2) C := k + 1 + 23−k . k 24 k 2 (2) 12Γ( 2 ) ζ (k)

In dimension k = 3 this becomes a double pole, and the Laurent series of D(s, Pk ×Pk) about s = 1 is given by

π2 π2(1 + γ − log(2π) − 24ζ0(−1) + 2ζ(2)(3)) 8a + + 0 . 3ζ(2)(3)(s − 1)2 3ζ(2)(3)(s − 1) s − 1

The function D(s, Pk × Pk) is otherwise analytic for Re s > 0 with the exception of a

1 single, simple pole at s = 2 in the case k = 4 with residue

0 C = Res D(s, P4 × P4) = Res D(s, S4 × S4), 4 1 3 s= 2 s=− 2

0 where C4 is deﬁned as in Remark 4.5.1.

4.6 Modiﬁcations in the Planar Case

As discussed at the start of this chapter, special techniques are required to study the

Dirichlet series D(s, Sk × Sk) and D(s, Pk × Pk) in the case k = 2. In this section, we show how the added diﬃculties of the k = 2 case may be addressed, proving in the

79 end a version of Theorem 4.5.2 for dimension k = 2.

To highlight the diﬀerences between the cases k = 2 and k ≥ 3, we discuss only those proofs that must adapt in the k = 2 case. In particular, since Propositions 4.1.1 and 4.1.2 hold for k = 2, we begin our discussion around §4.2.1, where we produced the spectral expansion for Zk(s, w).

4.6.1 Spectral Expansion of Z2(s, w)

We recall that

s+ k −1 k 2 k X Dk(s; h) (4π) 2 X h|θ (·)| Im(·) 2 ,Ph(·, s)i Z (s, w) = = . (4.33) k hw Γ(s + k − 1) hw h≥1 2 h≥1

from (4.6), which holds for all k ≥ 2 but requires slight modiﬁcation before the

Poincar´eseries Ph(z, s) can be replaced by its spectral expansion. In §4.2.2, this

k 2 k 2 modiﬁcation amounts to replacing |θ (·)| Im(·) 2 by the L (Γ0(4)\H) function

k k 2 2 k k Vk(z) := |θ (z)| Im(z) − E∞(z, 2 ) − E0(z, 2 ),

k which requires k ≥ 3 because Ea(z, s) has a pole at s = 2 when k = 2.

k 2 k In dimension k = 2, we instead mollify the behavior of |θ (z)| Im(z) 2 at cusps by subtracting the constant coeﬃcients of the Laurent series of these Eisenstein series about s = 1. This produces the following variant of Lemma 4.2.1.

Lemma 4.6.1. Deﬁne V2(z) by

2 2 V2(z) := |θ (z)| Im(z) − const E∞(z, s) − const E0(z, s), s=1 s=1

80 in which consts=c f(s) denotes the constant term of the Laurent expansion of f(s) at

2 s = c. Then V2(z) ∈ L (Γ0(4)\H).

Proof. We recall from the proof of Lemma 4.2.1 that

2 2 2 2 −2πy |θ (z)| Im(z) = |θ (σ0z)| Im(σ0z) = y 1 + O e

2 2 1 as y = Im z → ∞, and that |θ (z)| Im(z) vanishes at the cusp 2 . On the other hand, from the Fourier expansion (4.7) and the explicit computations (4.15), we ﬁnd that

log y 3 − 3 log π − 36ζ0(−1) − 13 log 2 const E∞(z, s) = y − + . s=1 2π 3π

Likewise, consts=1 E0(σ0z, s) = y + O(log y) as Im z → ∞, ie. as σ0z → 0. These constant terms otherwise vanish near the cusps of Γ0(4)\H, since the same was true for E∞(z, s) and E0(z, s). Since the constant terms of these Laurent expansions are

2 modular, we conclude that V2(z) ∈ L (Γ0(4)\H), as claimed.

2 2 Replacing |θ (z)| Im z with V2(z) and accounting for this change, we produce a variant of (4.33) for Z2(s, w) which is more amenable to spectral expansion:

s (4π) X hV2,Ph(·, s)i Z (s, w) = 2 Γ(s) hw h≥1 s (4π) X hconstu=1 E∞(·, u) + E0(·, u) ,Ph(·, s)i − . (4.34) Γ(s) hw h≥1

To better understand the correction term in (4.34), we write

const (E∞(z, u) + E0(z, u)) ,Ph(z, s) u=1 u+ 1 u− 1 2π 2 h 2 Γ(s + u − 1)Γ(s − u) = const ϕ∞h(u) + ϕ0h(u) s− 1 u=1 (4πh) 2 Γ(s)Γ(u) πΓ(s − 1) = ϕ (1) + ϕ (1), (4πh)s−1 ∞h 0h 81 in which we’ve used that the operator “const” commutes with the Petersson inner

product and that the coeﬃcients ϕah(s) are holomorphic at s = 1 for h ≥ 1. Thus

s 2 (4π) X hV2,Ph(·, s)i 4π X (ϕ∞h(1) + ϕ0h(1)) Z (s, w) = + . (4.35) 2 Γ(s) hw s − 1 hw+s−1 h≥1 h≥1

We now produce a spectral expansion for Z2(s, w) by substituting the spectral expansion of the Poincar´eseries (as given in (4.10)) into (4.35). Simpliﬁcation proceeds as in Proposition 4.2.4 and we conclude the following.

Theorem 4.6.2. The shifted convolution sum Z2(s, w) has a spectral expansion of the form

2 4π X (ϕ∞h(1) + ϕ0h(1)) Z (s, w) = (4.36) 2 s − 1 hw+s−1 h≥1

X 1 + 2π L(s + w − 2 , µj)G(s, itj)hV2, µji (4.37) j 1 +z 1 X Z G(s, z)π 2 + ζ (s + w, z)hV ,E (·, 1 − z)i dz, (4.38) i Γ( 1 + z) a 2 a 2 a (0) 2

in which G(s, z) is deﬁned as in Proposition 4.2.4 and ζa(s, z) is deﬁned in (4.16). As in Proposition 4.2.4, the three terms at right in (4.36)-(4.38) will be called the “non-spectral part,” the “discrete spectral part,” and the “continuous spectral part,” respectively.

4.6.2 Meromorphic Continuation of Z2(s, w)

In this section, we apply the spectral expansion of Z2(s, w) given in Theorem 4.6.2 to

2 produce a meromorphic continuation of Z2(s, w) to all of C . As in §4.3, our analysis

follows the decomposition of Z2(s, w) into the non-spectral part (4.36), the discrete spectral part (4.37), and the continuous spectral part (4.38).

82 The Non-Spectral Part

The non-spectral part (4.36) can be made wholly explicit in light of the equations (4.15). After a great deal of simpliﬁcation, we conclude that

2 4π X (ϕ∞h(1) + ϕ0h(1)) 8ζ(s + w − 1)ζ(s + w) = 1 − 21−s−w + 41−s−w . s − 1 hw+s−1 s − 1 h≥1

Thus the non-spectral part is meromorphic in C2, with polar lines at s + w = 2 and s + w = 1. Specializing to the case w = 0, we detect just two poles: a single pole at s = 2 and a double pole at s = 1.

The Discrete Spectral Part

The discrete spectral part (4.37) is handled exactly as in §4.3.2. We conﬁrm that this

2 1 term is meromorphic in C and analytic in Re s > 2 . In the distinguished case w = 0, 1 we note that additional cancellation implies that Z2(s, 0) is analytic in Re s > − 2 .

The Continuous Spectral Part

Likewise, the continuous spectral part (4.38) is handled exactly as in §4.3.3. In fact, the statements of Proposition 4.3.3, Lemma 4.3.4, and Lemma 4.3.5 as well as the explicit formulas for the residual pairs (4.18) and (4.19) hold, exactly as written, in the case k = 2.

4.6.3 Analytic Behavior of W2(s)

In this section, we apply the meromorphic continuation of Z2(s, 0) described in the

previous section to study the analytic properties of the function W2(s). Fortunately, many of the results we require can be taken, verbatim, from analogous statements in dimension k ≥ 3.

83 k+3 As in §4.4, we restrict our attention to the region Re s > − 2 , which represents 5 Re s > − 2 in the case k = 2. We recall that

2 X r2(m) W (s) = + 2Z (s + 2, 0). 2 ms+2 2 m≥1

Our discussion is guided by the decomposition of W2(s) into this ﬁrst term, which we call the diagonal term, as well as the three terms which comprise the spectral

expansion of Z2(s, 0). We discuss each separately and combine our observations to produce the following theorem.

Theorem 4.6.3. The function W2(s) is meromorphic in C and analytic in the right

half-plane Re s > 0. The rightmost pole of W2(s) occurs at s = 0 and is simple

2 5 with residue 2π . The function W2(s) is otherwise analytic in Re s > − 2 , with the 3 exception of a simple pole at s = − 2 with residue √ 3 2 3 2 8(4 − 2)ζ( 2 ) L( 2 , χ) Res W2(s) = ≈ 1.27046 77438. 3 2 s=− 2 7π ζ(3)

−1 Here, L(s, χ) denotes the L-function of the Dirichlet character χ = ( · ).

The Diagonal Part

The diagonal part of W2(s) may be understood in two ways: ﬁrstly, through the convenient closed formula

2 2 2 X r2(m) 16ζ(s) L(s, χ) = ms (1 + 2−s)ζ(2s) m≥1

84 due to Borwein and Choi [BC03]; and secondly, through Zagier regularization (as in section §4.4.1). Through Zagier regularization, we recognize that

(4π)shV ,E (·, s)i L(s, θ2 × θ2) 16ζ(s)2L(s, χ)2 2 0 = = , (4.39) Γ(s) ζ(2s) (1 + 2−s)ζ(2s)

initially in the region 0 < Re s < 1 and then extended through analytic continuation.

In particular, the diagonal term may be understood as an inner product against Vk not only in the case k ≥ 3 but in the case k = 2 as well.

The diagonal part is thus meromorphic in C and analytic in Re s > −1, with a double pole at s = −1 with principal part

4 32 0 48 0 4 8γ + log 2 + L (1, χ) − 2 ζ (2) + 3 π π . (s + 1)2 s + 1

Here, we have used that L(1, χ) = π/4 to simplify our expression. The diagonal

3 part has additional poles in the half-plane Re s < − 2 which arise from the zeros of (1 + 2−2−s)ζ(2s + 4) but is otherwise analytic.

Remark 4.6.4. In dimension k ≥ 3, the diagonal part admits poles at both s = −1

k and s = − 2 . In dimension k = 2, these poles collide and we obtain the double pole described above. This “collision” is particularly interesting given the cancellation

k ± that exists between the s = − 2 pole and T3/2 in dimension k ≥ 3. This cancellation persists in the planar case but now involves cancellation between double poles in the

± diagonal term, T3/2, and the non-spectral part.

Remark 4.6.5. As in dimension k ≥ 3, we will show that the diagonal part perfectly

3 cancels with a pair of residual terms in the half-plane Re s < − 2 . In particular, the poles of the diagonal part which arise from the denominator (1 + 2−s−2)ζ(2s + 4) will

not aﬀect our analysis of W2(s).

85 The Non-Spectral Part

The contribution towards W2(s) from the non-spectral part takes the form

16ζ(s + 1)ζ(s + 2) E (s) = 1 − 2−s−1 + 4−s−1 . (4.40) 2 s + 1

This term has a simple pole at s = 0 with residue 2π2 as well as a double pole at s = −1 with principal part

8 8(γ + log π) − − . (s + 1)2 s + 1

The non-spectral part is otherwise holomorphic.

The Discrete and Continuous Spectral Parts

5 In the region Re s > − 2 , the discrete spectral part is analytic and the only poles of ± the continuous spectral part may be read from the two residual pairs T3/2(s, 0) and ± T1/2(s, 0), in the half-planes in which they appear.

± Following (4.18) and (4.39), the contribution of T3/2 towards W2(s) may be written

3 3 s+ 2 + 4Γ(s + 2 )π hV2,E0(·, −s)i 4T 3 (s + 2, 0) = 2 2 Γ(s + 2) (4.41) s+3 2s+ 3 3 4 π 2 Γ(s + )Γ(−s) ζ(−s)2L(−s, χ)2 = 2 . Γ(s + 2)2 (1 + 2s)ζ(−2s)

+ Since T3/2(s + 2, 0) only appears in the meromorphic continuation of W2(s) in 1 the half-plane Re s < − 2 , we restrict our analysis of (4.41) to the vertical strip 5 1 Re s ∈ (− 2 , − 2 ). In this strip, there is a double pole at s = −1 with principal part

4 24 log π − 4 log 2 + 144ζ0(−2)/π2 32L0(1, χ) + − , (s + 1)2 3(s + 1) π(s + 1)

86 3 as well as a simple pole at s = − 2 with residue √ 8(4 − 2)ζ( 3 )2L( 3 , χ)2 2 2 . 7π2ζ(3)

3 The next pair of residual terms appears once Re s < − 2 and takes the form

s+2 2 2 + (4π) hV2,E∞(·, s + 2)i 16ζ(s + 2) L(s + 2, χ) 4T 1 (s + 2, 0) = − = − −s−2 2 Γ(s + 2) (1 + 2 )ζ(2s + 4)

in light of (4.19) and (4.39). Thus the second residual pair exactly cancels the diagonal

3 part in the half-plane Re s < − 2 , as claimed in Remark 4.6.5.

4.6.4 Polar Analysis of D(s, S2 × S2) and D(s, P2 × P2)

We now apply Theorem 4.6.3 to classify the rightmost poles of D(s, S2 × S2) and

D(s, P2 × P2). As this material simply echoes that of §4.5.1-4.5.2, we give a quick sketch of the method and follow up with a summary of our results, which we present in Tables 4.3 and 4.4.

The Rightmost Poles of D(s, S2 × S2)

To begin, we recall from Proposition 4.1.2 that

1 Z Γ(z)Γ(s + 2 − z) D(s, S2 × S2) = ζ(s + 2) + W2(s) + W2(s − z)ζ(z) dz, 2πi (σ) Γ(s + 2)

in which σ > 1 and Re s > σ to begin. We note that the integral is analytic in s for Re s > σ. Shifting (σ) to (−2 + ), we extract three residues and conclude that

W (s − 1) D(s, S × S ) = ζ(s + 2) + W (s) + 2 − 1 W (s) + 1 (s + 2)W (s + 1) 2 2 2 s + 1 2 2 12 2 1 Z Γ(z)Γ(s + 2 − z) + W2(s − z)ζ(z) dz. 2πi (−3+) Γ(s + 2)

87 Since the shifted integral is analytic for Re s > −2 + , we may read oﬀ the poles and

residues of D(s, S2 × S2) in this region by considering the other terms. In light of Theorem 4.6.3, this is quite easy. Our results are presented in Table 4.3.

3 Table 4.3: Summary of Polar Data for D(s, S2 × S2) in the Half-Plane Re s > − 2 pole location source of pole residue

E2(s−1) W2(s−1) 2 s = 1 s+1 , from s+1 π

1 1 2 s = 0 2 E2(s), from 2 W2(s) π √ 4T + (s+1,0) 3 2 3 2 1 3/2 W2(s−1) 16(4− 2)ζ( 2 ) L( 2 ,χ) s = − 2 s+1 , from s+1 7π2ζ(3)

s = −1 ζ(s + 2) 1

1 1 2 s = −1 12 (s + 2)E2(s + 1), from 12 (s + 2)W2(s + 1) π /6

W2(s−1) s = −1 s+1 W2(−2)

The Rightmost Poles of D(s, P2 × P2)

As in §4.5.2, our analysis of D(s, P2 × P2) begins with Proposition 4.1.1. In the case k = 2, this result takes the form

D(s, P2 × P2) = D(s − 2,S2 × S2) (4.42)

+ π2ζ(s − 2) − 2πζ(s − 1) − 2πL(s − 1, θk) (4.43) 2π Z Γ(z)Γ(s − 1 − z) − L(s − 1 − z, θ2)ζ(z) dz, (4.44) 2πi (σ) Γ(s − 1)

with σ > 1 and Re s > 1 + σ to begin. Thus the meromorphic properties of

3 1 D(s, S2 × S2) in Re s > − 2 determine those of D(s, P2 × P2) in Re s > 2 , once the correction terms in lines (4.43) and (4.44) are taken into account. Proceeding as in §4.5.2, we compute the poles and residues seen in Table 4.4.

88 1 Table 4.4: Summary of Polar Data for D(s, P2 × P2) in the Half-Plane Re s > 2 pole location line source of pole residue

E2(s−3) W2(s−3) 2 s = 3 (4.42) s−1 , from s−1 π s = 3 (4.43) π2ζ(s − 2) π2

L(s−2,θ2) 2 s = 3 (4.44) −2π s−2 −2π

1 1 2 s = 2 (4.42) 2 E2(s − 2), from 2 W2(s − 2) π s = 2 (4.43) −2πζ(s − 1) −2π s = 2 (4.43) −2πL(s − 1, θ2) −2π2 s = 2 (4.44) πL(s − 1, θ2) π2

L(s−2,θ2) 2 s = 2 (4.44) −2π s−2 −2πL(0, θ ) √ 4T + (s−1) 3 2 3 2 3 3/2 W2(s−3) 16(4− 2)ζ( 2 ) L( 2 ,χ) s = 2 (4.42) s−1 , from s−1 7π2ζ(3)

s = 1 (4.42) ζ(s), from D(s, S2 × S2) 1

W2(s−3) s = 1 (4.42) s−1 W2(−2)

sE2(s−1) sW2(s−1) π2 s = 1 (4.42) 12 , from 12 6

Remark 4.6.6. Note that most, but not all, of these residues can be computed by specializing the formulas given in Table 4.2 to the case k = 2. This highlights the similarity between the k = 2 and k ≥ 3 cases.

Cancellation in the Poles of D(s, P2 × P2)

We now demonstrate cancellation between the many polar terms in D(s, P2 × P2)

1 and classify the poles of D(s, P2 × P2) in the right half-plane Re s > 2 .

From Table 4.4, we see at once that the pole in D(s, P2 × P2) at s = 3 does not occur. In addition, the polar terms about s = 2 cancel oﬀ, as can be seen from

2 3 Table 4.4 and the calculation L(0, θ ) = −1 from (4.32). Thus the pole at s = 2

89 represents the rightmost pole of D(s, P2 × P2).

To understand the residue of the pole at s = 1, we must compute W2(−2). This

± computation is simpliﬁed by the fact that T1/2 cancels with the diagonal part of

W2(s) near s = −2, hence both terms may be ignored. The contribution from the ﬁrst residual pair can be read from (4.18) and vanishes due to zeros in 1/Γ(s + 2)2. Moreover, both the discrete spectral part (4.37) and the contour integral from the

continuous spectral part (4.38) of Z2(0, 0) vanish, because they include factors of

G(0, itj) = 0 and G(0, z) = 0, respectively. Thus

16ζ(−1)ζ(0) W (−2) = E (−2) = 1 − 22−1 + 42−1 = −2, 2 2 −1

2 by (4.40). We conclude that the pole of D(s, P2 × P2) at s = 1 has residue π /6 − 1.

This proves the following theorem, which extends the result of Theorem 4.5.2 to include the case k = 2.

Theorem 4.6.7. The function D(s, P2 × P2), originally deﬁned as a series in the

right half-plane Re s > 3, has a meromorphic continuation to C. This function is 3 3 analytic in the right half-plane Re s > 2 and has a pole at s = 2 with residue √ 16(4 − 2)ζ( 3 )2L( 3 , χ)2 C0 := 2 2 ≈ 2.54093 54876. 2 7π2ζ(3)

π2 The function D(s, P2 × P2) has a second pole at s = 1 with residue C2 := 6 − 1 and 1 is otherwise analytic in Re s > 2 .

Remark 4.6.8. In the end, the diﬀerences between our treatment of D(s, Pk × Pk) for k ≥ 3 in §4.1-4.5 and our treatment of D(s, P2 × P2) in the current section amount to little more than technical details. Thus we may use our results to

90 identify key diﬀerences between the behavior of D(s, Pk ×Pk) for k = 2 and for k ≥ 3.

3 In particular, we observe that the leading pole at s = 2 in dimension k = 2 5−k corresponds to a “travelling pole” at s = 2 in dimension k. This pole contributes to the rightmost pole at s = 1 in dimension k = 3 and is otherwise non-dominant. (For more about this pole in dimension k = 4, see Remark 4.5.1.)

Movement of this pole relative to a ﬁxed pole at s = 1 accounts for the apparent phase change in the size of the error terms in the generalized Gauss circle problem between dimensions k = 2 and k = 3 (as described in (1.1)). This phase change will be discussed in even greater abstraction in Chapter6.

91 Chapter 5

Arithmetic Results for the Generalized Gauss Circle Problem

In this chapter, we apply the results of Chapter4 to produce a variety of arithmetic results related to the Gauss circle problem and the generalized Gauss circle problem. To be speciﬁc, we consider the following arithmetic objects:

2 a. The Discrete Laplace Transform of Pk(n) : By the discrete Laplace transform, we refer to the smoothed sum

X 2 −n/X Pk(n) e . n≥1

This object was introduced and studied in the cusp form case in Theorem 3.5.4, and our treatment of it here is similar.

2 b. The (Continuous) Laplace Transform of Pk(n) : This object, also known

2 as the (ordinary) Laplace transform of Pk(n) , refers to the integral

Z ∞ 2 −t/X (Pk(t)) e dt. 0

92 2 The Laplace transform of Pk(t) was ﬁrst studied by Ivi´cin [Ivi01] in the case k = 2. We improve the error bounds given in [Ivi01] for dimension k = 2 and present new results in dimension k ≥ 3.

c. The Discrete Second Moment of Pk(n): By this, we refer to the ﬁnite sum

X 2 Pk(n) . n≤X

This object was studied brieﬂy in the cusp form case as a corollary (see Corol- lary 3.5.2) of the Wiener-Ikehara Theorem. We study it again now using cutoﬀ integral transforms.

d. The (Continuous) Second Moment of Pk(n): This object, also known as

the mean-square of Pk(n), refers to the integral

Z X 2 (Pk(t)) dt. 0

This object has been studied by numerous authors (especially in the classic case k = 2), so we restrict our exposition to the case k ≥ 3. For k = 3, we improve the result of Jarnik [Jar40] described in (1.2) and produce, for the ﬁrst time, a secondary main term and a power-savings error.

Our results in this chapter follow largely from Theorems 4.5.2 and 4.6.7, which classify the rightmost poles of D(s, Pk ×Pk) in the cases k ≥ 3 and k = 2, respectively.

We require as well some basic information about the growth of D(s, Pk ×Pk) in vertical strips (cf. Theorem 3.5.3), which we supply now.

Theorem 5.0.1. The function D(s, Pk × Pk) grows at most polynomially in | Im s| in vertical strips away from poles.

93 Proof. This proof of this Theorem is extremely similar to Theorem 3.5.3, its analogue in the cusp form case, so we sketch the general proof and focus on the diﬀerences.

As in the cusp form case, we ﬁrst bound Wk(s) in vertical strips.

Polynomial bounds for the diagonal and continuous parts of Wk(s) are treated exactly as in Theorem 3.5.3. Our bound for the non-spectral part of Wk(s) (as given in (4.23) for k ≥ 3 or (4.40) for k = 2) follows from convexity bounds for ζ(s) and Stirling’s approximation.

Our treatment of the discrete spectral part of Wk(s) is similar to that of Wf (s) when k is even but breaks down when k is odd and the underlying modular forms are of half-integral weight. For this reason, we eschew (3.26) in favor of the bounds

X 2 2k+3 −πT X 2 −πT 2 |hVk, µji| T e , |ρj(h)| e h T ,

T ≤|tj |≤2T T ≤|tj |≤2T which follow from [Kır15, Proposition 14] and [HHR16, (4.3)], respectively. The discrete spectrum is otherwise treated as in Theorem 3.5.3.

Since Wk(s) has a series expansion which is absolutely convergent (and therefore uniformly bounded) in Re s > 1, we may bound the Mellin-Barnes integral that appears in D(s, Sk × Sk) by shifting contours and estimating residues. It follows that D(s, Sk × Sk) is polynomially bounded in vertical strips, and the same holds for

D(s, Pk × Pk) by Proposition 4.1.1.

94 5.1 The Discrete Laplace Transform

2 We begin by considering the discrete Laplace transform of Pk(n) . As in §3.5, this is studied through the integral representation

s X X 1 Z X P (n)2e−n/X = P (n)2 Γ(s) ds k k 2πi n n≥1 n≥1 (σ) Z 1 s = D(s − k + 2,Pk × Pk)X Γ(s) ds, (5.1) 2πi (σ) in which σ 1 to begin. Since Γ(s) provides exponential decay in | Im s| in vertical

strips and D(s, Pk × Pk) grows at most polynomially in | Im s| by Theorem 5.0.1, we are justiﬁed in shifting the line of integration arbitrarily far to the left.

1 In dimension k = 2, we shift the line of integration to ( 2 +) for some small > 0, 3 extract residues at s = 2 and s = 1 (as described by Theorem 4.6.7), and bound the shifted integral in absolute value to produce the following theorem.

2 Theorem 5.1.1. The discrete Laplace transform of P2(n) satisﬁes

X 3 1 2 −n/X 0 3 2 2 + P2(n) e = C2Γ( 2 )X + C2X + O X n≥1

0 for all > 0, in which C2 and C2 are deﬁned as in Theorem 4.6.7.

Likewise, shifting the line of integration in (5.1) to (k − 2 + ) in dimension k ≥ 3 and extracting residues following Theorem 4.5.2 produces the following.

2 Theorem 5.1.2. For k ≥ 3, the discrete Laplace transform of Pk(n) satisﬁes

X 2 −n/X 0 k−1 k−1 Pk(n) e = δ[k=3]C3X (log X + 1 − γ) + CkΓ(k − 1)X n≥1

3 0 3 k− 2 k−2+ + δ[k=4]C4Γ(k − 2 )X + O X

95 0 0 for all > 0, in which the constants Ck, C3, and C4 are explicit constants.

0 0 Remark 5.1.3. The coeﬃcients C3, C4, and Ck (for k 6= 3) can be read from the

Laurent series coeﬃcients of D(s, Pk × Pk) about its rightmost poles and are given by the formulas

√ π2 16(9 2 − 8), ζ( 1 ζ( 3 )2ζ( 5 ) C0 = ,C0 = 2 2 2 , 3 3ζ(2)(3) 4 7π2ζ(3) k2v2 πkζ(k − 2) C = k + 1 + 23−k . k 24 k 2 (2) 12Γ( 2 ) ζ (k)

Note that the formula given for Ck holds for k = 2 even though our description of C2

was obtained in a diﬀerent way. The constant C3, which equals the coeﬃcient of the

−1 (s − 1) term of the Laurent expansion of D(s, P3 × P3) about the pole at s = 1, is

harder to describe explicitly. Numerical approximation suggests that C3 ≈ 10.6.

Remark 5.1.4. When k > 3 is small it is not diﬃcult to list the precise locations

3−k of the poles of D(s, Pk × Pk) in the half-plane Re s > 2 and produce additional main terms and improved error estimates in Theorem 5.1.2. For example, there exist

constants D4 and D5 such that

X 5 3 2 −n/X 3 0 5 2 2 2 + P4(n) e = 2C4X + C4Γ( 2 )X + D4X + O X , n≥1

X 2 −n/X 4 3 2+ P5(n) e = 6C5X + D5X + O X . n≥1

5.2 The Laplace Transform

2 Our second family of arithmetic results concerns the Laplace transform of Pk(n) . This transform is very similar to the discrete Laplace transform studied in the previous section and we obtain our results as corollaries of Theorems 5.1.1 and 5.1.2.

96 To begin, note that

k k k k Pk(t) = Sk(t) − vkt 2 = Sk(btc) − vkt 2 = Pk(btc) + vkbtc 2 − vkt 2 .

Squaring, it follows that

k k 2 k k 2 2 2 2 2 2 2 Pk(t) = Pk(btc) + vk btc − t + 2vkPk(btc) btc − t . (5.2)

2 Our analysis of the Laplace transform of Pk(t) follows the decomposition in (5.2). We approximate the Laplace transforms of each term and conclude by summing these approximations.

Lemma 5.2.1 (Laplace transform of the ﬁrst term in (5.2)). We have

∞ Z 3 2 −t/X 0 3 2 0 k−1 Pk(btc) e dt = δ[k=2]C2Γ( 2 )X + δ[k=3]C3X (log X + 1 − γ) 0

3 k−1 0 3 k− 2 ηk+ + CkΓ(k − 1)X + δ[k=4]C4Γ(k − 2 )X + O X ,

1 in which ηk = 2 for k = 2 and ηk = k − 2 for k ≥ 3.

Proof. We break the integral over [0, ∞) into intervals and directly compute

Z ∞ Z n+1 2 −t/X X 2 −t/X −1/X X 2 −n/X Pk(btc) e dt = Pk(n) e dt = X(1 − e ) Pk(n) e . 0 n≥0 n n≥0

2 The sum that remains is essentially the discrete Laplace transform of Pk(n) , and we produce asymptotics for this term by applying Theorems 5.1.1 and 5.1.2 and using the approximation X(1 − e1/X ) = 1 + O(1/X).

Remark 5.2.2. The constant ηk is just deﬁned to simplify the statement of our error

bounds. Note, however, that ηk = 2αk, where αk is the conjectured best upper bound in the k-dimensional generalized Gauss circle problem.

97 Lemma 5.2.3 (Laplace transform of the second term in (5.2)). We have

∞ 2 2 Z k k k v Γ(k − 1) 2 2 2 2 −t/X k k−1 k−2 vk (btc − t ) e dt = X + O(X ). 0 12

Proof. We break the integral into intervals and change variables to obtain

Z ∞ Z 1 2 k k 2 −t/X X −n/X k k −t/X (btc 2 − t 2 ) e dt = e (n + t) 2 − n 2 e dt. 0 n≥0 0

For k = 2, we integrate explicitly and approximate the series to prove the lemma. For k ≥ 3, note that the n = 0 term contributes O(1). For n ≥ 1, we factor out nk from the integrand and apply the binomial series to approximate the integrand. Thus

Z ∞ Z 1 2 2 3 k k 2 −t/X X k −n/X k t t −t/X (btc 2 − t 2 ) e dt = n e + O e dt + O(1). 4n2 k n3 0 n≥1 0

The main term in this approximation contributes

2 Z 1 2 k X k−2 − n 2 −t/X k 3 − 1 3 2 X k−2 − n n e X t e dt = 2X − e X (2X + 2X + X) n e X . 4 4 n≥1 0 n≥1

The sum that remains may be represented as a contour integral of the zeta function using the Cahen–Mellin integral from (3.23). By approximating the function of X outside the sum, shifting contours, and extracting residues, we obtain

k2 1 1 Z k2Γ(k − 1) + O ζ(s − k + 2)XsΓ(s) ds = Xk−1 + O Xk−2 . 12 X 2πi (k) 12

Similar techniques show that the error term in our binomial series propagates as

Z 1 X k−3 − n 3 −t/X X k−3 − n k−2 O n e X t e dt = O n e X = O X , n≥1 0 n≥1 which proves the lemma. (Note that this last estimate requires k ≥ 3.)

98 Lemma 5.2.4 (Laplace transform of the third term in (5.2)). We have

Z ∞ k k k t vkπ 2 kΓ(k − 1) 2 2 − X k−1 k−2+ 2vk Pk(btc) btc − t e dt = − k X + O X . 0 4Γ( 2 )

Proof. Arguing as in Lemma 5.2.3, we conclude that

Z ∞ k k − t Pk(btc) btc 2 − t 2 e X dt 0 Z 1 2 X k − n kt t − t = P (n)n 2 e X − + O e X dt + O(1). (5.3) k 2n k n2 n≥1 0

The main term in this approximation may be written

Z 1 k X k −1 − n − t − P (n)n 2 e X te X dt 2 k n≥1 0

k 2 − 1 2 X k −1 −n/X = − X − e X (X + X) P (n)n 2 e . (5.4) 2 k n≥1

P s To understand the sum that remains, let D(s, Pk) := n≥1 Pk(n)/n denote the

non-normalized Dirichlet series associated to Pk(n). Adapting (4.2) to account for

the diﬀerence between Sk(n) and Pk(n), we conclude that

k k k D(s, Pk) = ζ(s) + L(s − 2 + 1, θ ) − vkζ(s − 2 ) Z 1 k k Γ(z)Γ(s − z) + L(s − 2 + 1 − z, θ )ζ(z) dz, 2πi (σ) Γ(s)

k in which σ > 1. The poles and residues of D(s, Pk) in the right half-plane Re s > 2 −1 are summarized in Table 5.1.

99 k Table 5.1: Summary of Polar Data for D(s, Pk) in the Half-Plane Re s > 2 − 1 pole location source of pole residue

k k s = 2 + 1 vkζ(s − 2 ) vk k k k L(s− 2 ,θ ) s = 2 + 1 s−1 −vk

k k k k πk/2 s = 2 L(s − 2 + 1, θ ) 2 vk = k Γ( 2 ) k 1 k k k πk/2 s = 2 − 2 L(s − 2 + 1, θ ) − 4 vk = − k 2Γ( 2 )

s = 1 ζ(s) 1 k k L(s− 2 ,θ ) k k s = 1 s−1 L(1 − 2 , θ ) = −1

As in Lemma 5.2.3, the sum in (5.4) may be understood as an inverse Mellin transform. Doing so and shifting contours to extract residues, we conclude that

Z X k −1 − n 1 k s P (n)n 2 e X = D(s − + 1,P )X Γ(s) ds k 2πi 2 k n≥1 (k+1) k π 2 Γ(k − 1) k−1 k−2+ = k X + O,k X . 2Γ( 2 )

The main term from line (5.4) is therefore

k π 2 Γ(k − 1)k k−1 k−2+ − k X + O X . 8Γ( 2 )

Similarly, we see that error term in the integral in (5.3) propagates as

Z 1 X k −2 − n 2 −t/X X k −2 − n k−2+ O Pk(n)n 2 e X t e dt = O Pk(n)n 2 e X = O X , n≥1 0 n≥1 which completes the proof.

By combining the results of Lemmas 5.2.1, 5.2.3, and 5.2.4, we obtain an asymp-

2 totic for the Laplace transform of Pk(t) . For readability’s sake, we split our result into three cases: k = 2, k = 3, and k ≥ 4. 100 2 Theorem 5.2.5. For all > 0, the Laplace transform of Pk(t) satisﬁes

∞ Z 3 1 2 −t/X 0 3 2 2 + k = 2 : (P2(t)) e dt = C2Γ( 2 )X − X + O X 0 Z ∞ 2 −t/X 0 2 2π2 2 1+ k = 3 : (P3(t)) e dt = C3X (log X + 1 − γ) + C3 − 3 X + O X 0 Z ∞ Γ(k − 1)πk k ≥ 4 : (P (t))2e−t/X dt = C Γ(k − 1)Xk−1 − Xk−1 k k k 2 0 6Γ( 2 )

5 0 5 2 k−2+ + δ[k=4]C4Γ( 2 )X + O X

Remark 5.2.6. We are not the ﬁrst to study the Laplace transforms of the functions

2 2 Pk(t) . In [Ivi01], Ivi´cstudied the Laplace transform of P2(t) and proved that

∞ Z 3 2 2 −t/X 0 3 2 3 + P2(t) e dt = C2Γ( 2 )X − X + O X 0 for all > 0. Theorem 5.2.5 improves Ivi´c’s error bound and generalizes his result to arbitrary dimension.

As with the discrete Laplace transform, it is possible to reduce the error bounds

k−1 + in Theorem 5.2.5 to O(X 2 ) in dimension k by extracting additional main terms. This would strengthen Theorem 5.2.5 in dimension k ≥ 4.

5.3 The Discrete Second Moment

In this section, we derive asymptotic expressions for the discrete second moment of the

2 lattice point discrepancies Pk(n) . It does not appear that results of this particular form have appeared in the literature before, but this likely due to a general preference of others towards continuous second moments, which we treat in the following section.

101 We focus on the cases k ≥ 3 in this section and the next. This is not due to some mysterious failure in the case k = 2; rather, this choice has been made to reduce casework in our exposition and because the results one obtains this way in dimension k = 2 are previously known.

To produce a second moment result without smoothing, we introduce two additional smooth cutoff transforms and study them in tandem. The first of these, which we call uy(x), represents a generic smooth cutoff of compact support (with optimiz- ing parameter y). The precise definition and key properties of uy(x) are given in the following proposition.

+ Proposition 5.3.1. Let uy(x): R → [0, 1] be a smooth function satisfying

a. uy(x) = 1 for x ∈ [0, 1],

b. uy(x) is supported on [0, 1 + 1/y].

Then the Mellin transform of uy(x), which we denote by Uy(s), satisﬁes:

1. Uy(s) = 1/s + Os(1/y),

0 2 2. Uy(s) = −1/s + Os(1/y),

1 y m 3. For all m ≥ 1, Uy(s) y 1+| Im s| .

Proof. Statement (1) follows from the deﬁnition of uy(x), since

Z ∞ Z 1 Z 1+1/y s dt s−1 s dt 1 1 Uy(s) = uy(t)t = t dt + uy(t)t = + Os . 0 t 0 1 t s y

For (2), diﬀerentiation under the integral sign yields

Z 1 Z 1+1/y 0 s dt s dt 1 1 Uy(s) = t log t + uy(t)t log t = − 2 + Os . 0 t 1 t s y

102 Repeated integration by parts on the definition of Uy(s) gives (3) for integers m ≥ 1, and the general result follows from the Phragmén–Lindelöfprinciple.

The second cutoﬀ transform we require is an integral transform that concentrates

mass in the region where uy(s) decreases to 0, namely

Z 2 2 2 1 s πs ds y log X X exp 2 = exp − . (5.5) i (σ) y y 4π

This integral identity may be veriﬁed by writing Xs = exp(s log X), completing the square in s, and comparing the result to the standard Gaussian. Note that the transform (5.5) experiences exponential decay outside of the range X ∈ [1 − 1/y, 1 + 1/y].

2 As the next proposition shows, one can produce sharp estimates for Pk(n) (and in general, for any sequence with non-negative coeﬃcients), by combining these two cutoﬀ transforms.

Proposition 5.3.2. We have

X 1 Z P (n)2 = D(s − k + 2,P × P )XsU (s) ds k 2πi k k y n≤X (k) Z 2 1 s πs ds + O D(s − k + 2,Pk × Pk)X exp 2 . i (k) y y

Proof. With σ 1 large enough to ensure absolute convergence, linearity and Mellin inversion suﬃce to give

s 1 Z 1 X Z X D(s − k + 2,P × P )XsU (s) ds = P (n)2 U (s) ds 2πi k k y 2πi k n y (σ) n≥1 (σ) X n X X n = P (n)2 u = P (n)2 + P (n)2 u . (5.6) k y X k k y X n≥1 n≤X X

103 To bound the rightmost sum in (5.6), we ﬁrst remark that

y2 log2(X/n) y2 log2(1 + 1/y) exp − exp − 1, 4π 4π

2 uniformly in n, for n ∈ [X,X + X/y]. Since Pk(n) ≥ 0 and uy(x) ≤ 1, it follows that

X n X y2 log2(X/n) P (n)2u P (n)2 exp − . (5.7) k y X k 4π X

Our result now follows by extending the range of summation at right in (5.7) to the range n ≥ 1 and interpreting the result via (5.5).

To understand the discrete second moment it therefore suﬃces to understand the two integral transformations which appear in Proposition 5.3.2. We begin with uy(x), the smooth cutoﬀ of compact support.

Lemma 5.3.3. For each k ≥ 3, there exists M > 0 such that

Z 1 s D(s − k + 2,Pk × Pk)X Uy(s) ds 2πi (k) 0 2 k−1 k−1 C3X 1 CkX X log X k− 5 M = δ log X − + + O + X 4 y . [k=3] 2 2 k − 1 y

1 Proof. Shifting the line of integration to (k − 1 − 4 ) passes a pole at s = k − 1. In the case k 6= 3, this pole is simple with residue

C Xk−1 Xk−1 C Xk−1U (k − 1) = k + O . k y k − 1 y

When k = 3, the pole at s = k − 1 is a double pole and contributes the residue

0 2 0 0 2 C3Uy(2)X log X + C3Uy(2) + C3Uy(2) X C0 X2 C C0 X2 log X = 3 X2 log X + 3 − 3 X2 + O . 2 2 4 y 104 M−1 By Theorem 5.0.1, there exists M > 0 such that D(s, Pk × Pk) | Im s| on

1 the line Re s = (k − 1 − 4 ). Bounding the shifted integral by

k−1− 1 Z ∞ M+1 X 4 M−1 y k− 5 M (1 + |t|) dt = O X 4 y y −∞ 1 + |t|

via Proposition 5.3.1(3) completes the proof.

A similar argument bounds the concentrating integral transform, and with it, the error term in Proposition 5.3.2.

Lemma 5.3.4. For each k ≥ 3, there exists M > 0 such that

2 k−1 1 Z πs ds X log X 5 s k− 4 M D(s − k + 2,Pk × Pk)X exp 2 = O + X y . i (k) y y y

5 Proof. As in Lemma 5.3.3, we shift the line of integration to (k − 4 ) and extract a residue which is O(Xk−1 log X/y). To bound the shifted integral, choose M such that

M 5 D(s, Pk × Pk) | Im s| on the line Re s = k − 4 . The shifted integral is then

k− 5 ∞ 2 X 4 Z πt 5 M k− 4 M O (1 + |t|) exp − 2 dt = O X y , y −∞ y

as claimed.

Applying Lemma 5.3.3 and Lemma 5.3.4 in the context of Proposition 5.3.2 gives

2 an asymptotic expression for the discrete second moment of Pk(n) with an unspeciﬁed power-savings error term.

Theorem 5.3.5. For each k ≥ 3 there exists a λ > 0 (dependent on k) such that

0 0 X C C Ck P (n)2 = δ Xk−1 3 log X − 3 + Xk−1 + O Xk−1−λ . k [k=3] 2 4 k − 1 λ n≤X

105 Proof. The error terms in Lemma 5.3.3 and Lemma 5.3.4 are minimized in the case yM+1 = X1/4. We obtain the result by setting λ < (4(M + 1))−1.

Remark 5.3.6. Theorem 5.3.5 will be applied in the next section to derive asymptotics for the mean-square of Pk(t). Despite the unspeciﬁed power-savings we produce, the result we obtain in dimension k = 3 already improves theorems of Jarnik [Jar40] and Lau [Lau99]. We are led to wonder if our k = 3 result can be made even stronger.

In AppendixA we prove an explicit version of Theorem 5.0.1 in the case k = 3 and thereby obtain versions of Lemmas 5.3.3 and 5.3.4 with an explicit M. To be exact, we prove in Lemma A.2.2 that

7 D(σ + it, P3 × P3) (1 + |t|) 2

1 7 for all σ ∈ (0, 2 ). By shifting the lines of integration to (1 + ) instead of ( 4 ) in the proofs of Lemmas 5.3.3 and 5.3.4, we prove the explicit bounds

2 2 X log X 1+ 9 X log X 1+ 7 O + X y 2 ,O + X y 2 . y y

Balancing errors by choosing y = X2/11, we obtain the following explicit result:

Theorem 5.3.7. For all > 0, we have

0 0 X 2 C3 2 C3 C3 2 2− 2 + P (n) = X log X + − X + O X 11 . 3 2 2 4 n≤X

106 5.4 The Second Moment

In our ﬁnal arithmetic application, we study the (continuous) second moment of the

2 discrepancies Pk(n) ; namely, the integrals

Z X 2 (Pk(t)) dt. 0

Of the four types of averages we discuss in this chapter, the second moment has been studied the most extensively. Asymptotics for the second moment appear, with incremental improvements, in the work of Landau [Lan20], Cramer [Cra22], Walﬁsz [Wal27], Jarnik [Jar40], Chandrasekhran–Narasimhan [CN64], Kat´ai[K65´ ], Lau [Lau99], Nowak [Now04], and many others.

Just as Theorem 5.2.5 was obtained as an application of Theorems 5.1.1 and 5.1.2, we derive our results on the second moment as an application of Theorem 5.3.5. Again,

2 our proof follows the decomposition of Pk(t) into three terms following identity (5.2),

k k 2 k k 2 2 2 2 2 2 2 Pk(t) = Pk(btc) + vk btc − t + 2vkPk(btc) btc − t , (5.8)

and we compute the second moment of each term at right in a series of lemmas.

Lemma 5.4.1 (Second moment of the ﬁrst term in (5.8)). For each k ≥ 3, there exists λ > 0 such that

Z X 0 0 2 k−1 C3 C3 Ck k−1 k−1−λ Pk(btc) dt = δ[k=3]X log X − + X + Oλ X . 0 2 4 k − 1

2 Proof. When X is an integer, the integral of Pk(btc) is essentially the discrete second moment. When X is non-integral, the additional contribution of the integral over

2 [bXc,X] can be bounded using pointwise estimates for Pk(n) .

107 Lemma 5.4.2 (Second moment of the second term in (5.8)). We have

X 2 2 Z k k 2 k v 2 2 2 k k−1 k−2 vk btc − t dt = X + O X . 0 12(k − 1)

Proof. Our proof is similar to that of Lemma 5.2.3. We compute

Z bXc bXc Z 1 2 2 3 k k 2 X k k t t btc 2 − t 2 dt = n + O dt + O(1) 4n2 k n3 0 n=1 0 k2 X X k2Xk−1 = nk−2 + O nk−3 = + O(Xk−1) 12 12(k − 1) n≤bXc n≤X

by approximating the binomial series. The contribution towards the second moment of this term from bXc to X is O(Xk−2), which is absorbed into the error term.

Our calculation of the second moment of the third term in (5.8) resembles the proof of Lemma 5.2.4 but requires a technical result concerning the average order of

Pk(n). We give a crude estimate that is suﬃcient for our applications and discuss potential reﬁnements after the proof.

Lemma 5.4.3. We have

k k−1 X k −1 π 2 X γ + P (n)n 2 = + O X k k 2(k − 1)Γ( k ) n≤X 2

547 155 4 for all > 0, in which γ2 = 1248 , γ3 = 96 , and γk = k − 3 for k ≥ 4.

Proof. To estimate these partial sums we apply uy(x), the smooth cutoﬀ of compact

support introduced in Proposition 5.3.1. With D(s, Pk) as in Lemma 5.2.4, we have

Z X k −1 1 k s P (n)n 2 = D(s − + 1,P )X U (s) ds (5.9) k 2πi 2 k y n≤X (k) X+X/y X k −1 n + O |P (n)|n 2 u . (5.10) k y X n=X+1

108 k βk+ + The error term in (5.10) is O(X 2 /y), where βk denotes the smallest ex-

βk+ 131 ponent for which Pk(X) = O(X ) is currently known. Thus β2 = 416 following 21 k Huxley [Hux03], β3 = 32 following Heath-Brown [HB99], and βk = 2 − 1 in the case k ≥ 4 classically.

We estimate the integral in (5.9) by shifting the contour to (k −2+2). According to Table 5.1, this passes a single pole at s = k − 1 with residue

k k k−1 k−1 π 2 U (k − 1) π 2 X X y Xk−1 = + O . k k y 2Γ( 2 ) 2(k − 1)Γ( 2 )

To estimate the shifted contour integral, we ﬁrst bound the contributions of the various terms in the decomposition

L(s − k , θ2) D(s, P ) = ζ(s) + 1 L(s − k + 1, θk) − v ζ(s − k ) + 2 (5.11) k 2 2 k 2 s − 1 Z 1 k k Γ(z)Γ(s − z) + L(s − 2 + 1 − z, θ )ζ(z) dz (5.12) 2πi (−1+) Γ(s)

k on the line Re s = 2 − 1 + 2. In this region, the L-function in (5.12) is evaluated in its right half-plane of convergence and is uniformly bounded. Using the functional

z πz equation 2ζ(z)Γ(z) = (2π) ζ(1 − z) sec( 2 ) and Stirling’s approximation, we bound the integrand in (5.12) by

k − 1 + (1 + |s − z|) 2 2 − π (|z|+|s−z|−|s−z|) e 2 . k − 3 +2 (1 + |s|) 2 2

For |z| < |s|, the exponential term is O(1) and the integrand is O((1 + |s|)1−) on an interval of length O(|s|). This range then contributes O((1 + |s|)2−) after integration. Exponential damping in the range |z| > |s| creates a second contribution of size O((1 + |s|)2−), and we bound the integral by O((1 + |s|)2−) overall.

109 We bound the various L-functions in (5.11) using functional equations and the Phragm´en–Lindel¨of convexity principle. Given the bounds

1 1 3 1 4 2 2 ζ( 2 + 2 + it) (1 + |t|) , ζ(2 + it) (1 + |t|) , ζ(−1 + 2 + it) (1 + |t|) ,

k 1 k 5 L(2 + it, θ ) (1 + |t|) ,L(−1 + 2 + it, θ ) (1 + |t|) 2 ,

k 2− and our estimate for the integral (5.12), we conclude that D(s− 2 +1,Pk) (1+|s|) k on the line Re s = 2 − 1 + 2. Thus

Z k−2+2 Z ∞ 3 1 k s X 2− y D(s − 2 + 1,Pk)X Uy(s) ds (1 + |t|) dt 2πi (k−2+2) y −∞ 1 + |t|

by Proposition 5.3.1(3), and it follows that

k k ! k−1 βk+ + k−1 X k −1 π 2 X X 2 X 2 k−2+2 P (n)n 2 = + O + + y X . k 2(k − 1)Γ( k ) y y n≤X 2

Lower bounds on βk imply that the first error bound will always dominate the second. We balance the first and third error terms by fixing y as a power of X, and our stated estimates for βk give the lemma.

Remark 5.4.4. It may surprise the reader to see explicit bounds in Lemma 5.4.5, given that explicit bounds are not even stated in Lemma 5.4.1. This precision is a necessary evil which guarantees that current partial progress towards the generalized Gauss circle problem gives error terms no larger than O(Xk−1−λ), for some λ > 0.

It may be possible to improve these estimates by recognizing the partial sum of

Pk(n) in terms of the ﬁrst Riesz mean of Sk(n). For the deﬁnition of Riesz means and a brief summary of their properties, see Chapter7.

We now return to the proof of our second moment result.

110 Lemma 5.4.5 (Second moment of the third term in (5.8)). We have

Z X k k−1 k k π 2 kvkX 2 2 γk+ 2vk Pk(btc) btc − t dt = − k + O X 0 4(k − 1)Γ( 2 )

for all > 0, where γk is deﬁned as in Lemma 5.4.3.

Proof. Familiar approximations using the binomial series yield

Z bXc bXc Z 1 2 k k X k kt t P (btc) btc 2 − t 2 dt = P (n)n 2 − + O dt + O(1) k k 2n k n2 0 n=1 0 bXc k X k −1 X k −2 = − P (n)n 2 + O P (n)n 2 . 4 k k k n=1 n≤X

We estimate the ﬁrst term in the line above with Lemma 5.4.3 and bound the error

βk+ term using the estimate Pk(X) = O(X ). The contribution of the second moment

β + k −1 from the interval [bXc,X] is O(X k 2 ), which is absorbed into an existing error.

By combining Lemma 5.4.1, Lemma 5.4.2, and Lemma 5.4.5, we produce the

following theorem, our second moment estimate for Pk(n).

Theorem 5.4.6. For each k ≥ 3, there exists λ > 0 (dependent on k) such that

Z X C0 C0 πk Xk−1 P (t)2dt = δ Xk−1 3 log X − 3 + C − + O Xk−1−λ . k [k=3] 2 4 k k 2 k − 1 λ 0 6Γ( 2 )

Remark 5.4.7. In dimension k = 3, Theorem 5.4.6 represents an improvement over the results of Jarnik [Jar40] and Lau [Lau99], who prove

  √  Z X 0 X2 log X, [Jar40] 2 C3   Pk(t) dt = log X + O   . 2 0  2  X , [Lau99]

In particular, Theorem 5.4.6 is the ﬁrst result of its kind to identify the secondary main term of size X2 and to provide an error with power-savings. 111 Remark 5.4.8. The error bounds in Theorem 5.4.6 depend on the error bounds in Lemma 5.4.1, which, in turn, depend on those in Theorem 5.3.5. Second moment results with explicit power-savings thus follow from explicit bounds in the discrete second moment.

By applying the explicit version of our discrete second moment result in dimension k = 3 (Theorem 5.3.7) we obtain a version of Theorem 5.4.6 with explicit power- savings.

Theorem 5.4.9. For all > 0, we have

Z X 0 0 2 2 C3 2 C3 C3 π 2 2− 2 + P3(t) dt = X log X + − − X + O X 11 . 0 2 2 4 3

112 Chapter 6

A Brief Note on Sums of Coeﬃcients of Modular Forms

In previous chapters we have studied examples of modular forms f(z) = P a(n)e(nz) (both cuspidal and non-cuspidal) for which the average order

X Sf (X) := a(n) n≤X may be approximated with an error term which is remarkably small.

In the cusp form case from Chapter3, we see compelling evidence for the con-

k−1 + 1 + jecture that weight k cusp forms satisfy Sf (X) X 2 4 . The generalized Gauss circle problem from Chapters4–5 describes a more nuanced conjecture; namely, that

   1 + X 4 , k = 2 X k   Sk(X) := rk(n) = vkX 2 + O   .  k −1+  n≤X X 2 , k ≥ 3

The apparent phase change between dimensions two and three (ie. between

3 modular forms of weight 1 and weight 2 ) is noted in Remark 4.6.8, where it is

113 understood in terms of a shift in the relative locations of two potentially dominant singularities of a Dirichlet series.

In this chapter, we explain the cause of this phase change in a more general setting.

6.1 Sums of Coeﬃcients of Eisenstein Series

The ﬁrst examples of modular forms in an introductory course are often the (holo-

morphic) Eisenstein series Ek(z), which, for k ≥ 4 even, are modular forms of level 1 and weight k. As modular forms, these admit Fourier expansions, which here have the explicit form

∞ X (2πi)k X E (z) = a(n)e(nz) = 1 + σ (n)e(nz). k (k − 1)! ζ(k) k−1 n≥0 n=1

The explicit nature of these coeﬃcients makes it easy to compute their partial

sums. Applying well-known estimates for the average order of σν(n) (see [Apo76, Theorem 3.5], e.g.) gives the following proposition.

Proposition 6.1.1. For k ≥ 4 even, we have

(2πi)k X (2πi)k S (X) := 1 + σ (n) = Xk + O Xk−1 . Ek (k − 1)! ζ(k) k−1 k! n≤X

Note that SEk (X) demonstrates the sort of separation between main and error terms that we expect in the generalized Gauss circle problem in dimensions k ≥ 3. As the following proposition shows, this is the expected behavior not only for Eisenstein series but for all non-cuspidal modular forms of level 1.

114 Proposition 6.1.2. Let f(z) = P a(n)e(nz) be a modular form of level 1 and integral weight k. Then (2πi)k a(0) S (X) = Xk + O Xk−1 . f k!

Proof. In level 1, we may write f(z) = a(0)Ek(z) + g(z), where g(z) is a cusp form.

k−1 + 1 By linearity and the Hafner–Ivi´cbound Sg(X) X 2 3 [HI89], it follows that

k (2πi) a(0) k k−1 k−1 + 1 S (X) = a(0)S (X) + S (X) = X + O X + X 2 3 . f Ek g k!

The ﬁrst error dominates unless k ≤ 5/3, which completes the proof since k ≥ 4.

k−1 + 1 + Remark 6.1.3. Under the stronger hypothesis Sg(X) X 2 4 from the cusp

k−1 3 form analogy, one has that X dominates for all weights k ≥ 2 , which coincides with the phase change in the generalized Gauss circle problem.

6.2 Weight One Eisenstein Series

The limited results of the previous section suggest that the only modular forms f(z)

k−1+ for which the error term in the estimate for Sf (X) cannot satisfy O(X ) for all > 0 are either:

1. cusp forms

3 2. modular forms of weight k < 2 .

k−1 + 1 + In these cases we have the conjectural error bound O(X 2 4 ).

A particularly nice class of modular forms for which (2) holds are the Eisenstein series of weight one. We will not describe their analytic properties in detail here as our comments will be brief. For a full exposition, we direct the reader to [Miy06, §7.2].

115 Following Miyake (but also the note [Buz12]), let χ be an even Dirichlet character of conductor N and let ψ be an odd Dirichlet character of conductor M. Then the function f(z) with Fourier expansion P a(n)e(nz) deﬁned by

  1 L(0, ψ),N = 1 X n  2 a(n) = ψ(d)χ , a(0) = d  d|n 0, else is a weight one Eisenstein series of level MN, and each weight one Eisenstein series of level MN arises in this way.

With χ = 1 and ψ the primitive character of conductor 4, one obtains

X X d−1 2 1 a(n) = ψ(d) = (−1) = 4 r2(n) d|n d|n, d odd

following a classic formula of Jacobi. In other words, one may generalize the Gauss circle into the following question: what bounds hold for the error terms in partial sums of coeﬃcients of weight one Eisenstein series?

Remark 6.2.1. When χ is trivial (and sometimes in general), the coeﬃcients a(n) are called twisted divisor sums. These sums should remind the reader of the classic divisor sum d(n) and, in the context of average orders, the Dirichlet Divisor Problem:

Question (The Dirichlet Divisor Problem). What is the minimal exponent α for which X d(n) = X log X + (2γ − 1)X + O Xα+ n≤X for all > 0?

1 Dirichlet established α ≤ 2 as a consequence of his hyperbola method. We have 131 α ≤ 416 ≈ 0.3149 by a result of Huxley [Hux03] and it is widely conjectured that

116 1 α = 4 on the basis of Ω± and moment results. In these three senses (initial bounds, current progress, and conjecture), the Dirichlet divisor problem mirrors the Gauss circle problem. For more background on the divisor problem and its relation to the circle problem, we refer the reader to the survey article [IKKN06].

6.3 Conjectures for the Eisenstein Series Analogy

In this section, we state a conjecture for the partial sums of weight one Eisenstein series and illustrate it with a numerical example. To motivate the precise shape of the conjecture that follows, we ﬁrst compute the main term (if it exists) in the asymptotic of our partial sum. Our preliminary error bound is crude but matches the ﬁrst error bounds given by Gauss and Dirichlet.

Proposition 6.3.1. Let ψ and χ be as before. For all > 0, we have

X X n 1 + S (X) := ψ(d)χ = δ L(1, ψ)X + O X 2 . ψ,χ d [N=1] n≤X d|n

Proof. The divisor sum may be written as the Dirichlet convolution ψ ? χ, which has generating function L(s, ψ)L(s, χ). Thus

1 Z X n S (X) = L(s, ψ)L(s, ψ)XsU (s) ds + O d(n)u ψ,χ 2πi y y X (2) n

in which we’ve used the residue theorem and the fact that Dirichlet L-functions are entire unless they have trivial conductor. The contour integral along () is O(Xy), as follows from the convexity estimate L( + it, ψ)L( + it, χ) (1 + |t|)1− and √ Proposition 5.3.1(3). Choosing y = X balances errors and gives the result.

117 As in the circle and divisor problems, improved error estimates may be obtained by Voronoi summation (see [Coh07, §10.2.5] for a general discussion along these lines). It seems reasonable to conjecture the following, which we call the Eisenstein series analogy to the circle and divisor problems:

Conjecture 6.3.2 (The Eisenstein Series Analogy). Let ψ and χ be as before. Then

α 1 inf α : Sψ,χ(X) = δ[N=1]L(1, ψ)X + O(X ) = 4 .

We are of course unable to prove this conjecture. But, to at least give it some credibility, we illustrate it with an example.

Example 6.3.3. We consider the weight one Eisenstein series of level 7 deﬁned by

· the choices χ = 1 and ψ = 7 . In this case, Conjecture 6.3.2 predicts that

1 + π Sψ,χ(X) = CX + O X 4 ,C = √ ≈ 1.18741 7

1 6 The values of the normalized diﬀerence (Sψ,χ(X)−CX)/X 4 for integers X ≤ 10 are depicted in Figure 6.3. Note that the values appear to be bounded by O(X) for all > 0.

118 119

1 6 Figure 6.1: The normalized error terms (Sψ,χ(X) − CX)/X 4 , for X ≤ 10 . Chapter 7

Iterated Partial Sums

This chapter addresses a few questions that arise in the consideration of iterated partial sums of Fourier coeﬃcients of modular forms. Iterated partial sums, also known as C´esaro means, and their cousins, Riesz means, are important tools in summability theory and have a rich history in relation to the Gauss circle problem. In this chapter, we discuss these means and some of their applications in the Gauss circle problem and the cusp form analogy.

7.1 C´esaro and Riesz Means

To any coefficient series {a(n)}, we define the associated αth Césaro mean Sα(n) by setting S−1(n) = a(n) and applying the inductive relation

X Sα(n) := Sα−1(m). m≤n

120 In sums of this form an individual term a(n) appears many times. Reordering the sums to collect common terms, we compute

X n − m + α Sα(n) = a(m). (7.1) α m≤n

Thus Sα(n) may be viewed as a polynomial re-weighting of the ordinary partial sum S0(n). Note as well that (7.1) may be used to deﬁne Sα(n) for non-integral α.

Similarly, we deﬁne the αth Riesz mean Rα(n) by the identity1

X Rα(n) = (n − m)αa(m). m≤n

The Riesz mean is particularly useful in analytic number theory due to the convenient integral representation

  α Xα Z Γ(z)Γ(α + 1) X z (X − n) , n ≤ X dz = 2πi (σ) Γ(z + α + 1) n  0, else,

initially valid for α > 0 and σ > 0 [Apo76, §13.2, Lemma 3]. In the case α = 0, we recover Perron’s formula if the integral is considered in the sense of Cauchy’s principal value. Similar formulas exist for C´esaromeans but, like Perron’s formula, require regularization to converge.

1Some authors will normalize the Riesz mean by dividing through by nα. We opt to keep this factor because it creates a better analogy with the C´esaro mean and because it makes it easier to cross-reference certain formulas of Hardy and Jutila.

121 7.2 Relation to the Gauss Circle Problem and the

Cusp Form Analogy

Riesz means serve a critical role in Hardy’s proof [Har17] that the conjectured error bounds in the circle and divisor problems hold on average. To describe his results,

α let R2 (n) denote the αth Riesz mean for r2(n). Then Hardy [Har17, (2.41)] showed

α+1 √ α πX α Γ(α + 1) α + 1 X r2(n) R (X) = − X + X 2 2 J 2π nX (7.2) 2 α + 1 πα n(α+1)/2 α+1 n≥1

for all α ≥ 0, in which Jν(z) denotes the J-Bessel function. Hardy then shows that

Z X 2 α X α 3 +α R (t) − π (t − n) dt = O X 2 2 0 n≤t

for α > 0, and its validity in the case α = 0 (ie. the result for the partial sum P2(t)) is obtained using a limiting procedure.

Remark 7.2.1. In general, the series (7.2) is only conditionally convergent. Known asymptotics for the J-Bessel function show that this convergence is absolute once

1 α > 2 . In that range, we have

α+1 α πX α α + 1 R (X) = − X + O X 2 4 . 2 α + 1

α This estimate then implies sharp bounds for S2 (X), the αth C´esaromean for r2(n). When α is integral, this follows at once from the fact that {(X − z)j} is a basis for

C[z]. We obtain, for example, that

3 1 1 0 π 2 4 S2 (X) = R2(X) + R2(X) = 2 X − X + O X + πX + O (P2(X))

3 π 2 4 = 2 X + (π − 1)X + O X .

122 This error term can be improved to O(P2(X)) if one accounts for the value of the

1 convergent series (7.2). Note that S2 (X) counts the Z-lattice points in the paraboloid x2 + y2 ≤ z ≤ X and that the main term πX2/2 represents the volume of this region.

Riesz means also appear in the work of Jutila on sums of Fourier coeﬃcients of cusp forms [Jut87]. There, Jutila extends the Voronoi formulas used in the circle P problem to bound Sf (X), where f(z) = a(n)e(nz) is a weight k cusp form on

SL2(Z). As in Hardy’s work, it is helpful to introduce the αth Riesz mean of a(n),

α which we denote Rf (X). Jutila [Jut87, (1.6.5)] proves the following variant of (7.2):

k/2 ∞ √ α (−1) k + α X a(n) R (X) = X 2 2 J 4π nX . (7.3) f 2π n(k+α)/2 k+α n=1

1 As with (7.2), this series converges absolutely only for α > 2 . For such α, we obtain

k−1 1 α α 2 + 4 + 2 Rf (X) = O X , (7.4)

α which implies a matching bound for Sf (X), the αth C´esaromean of {a(n)}. Thus every partial sum after the ﬁrst experiences another round of square-root cancellation.

Remark 7.2.2. Jutila actually considers Voronoi formulas for the additively twisted coeﬃcients a(n)e(k/n). Unfortunately, what had once been a cusp form of weight κ and a twist of period k has become a blob of k’s at the hand of a less-than-zealous typist. This ambiguity aﬀects all of [Jut87] and provides the eager reader with an endless supply of “exercises.”

As we show in the following proposition, the bound (7.4) is not only an upper

α bound but the true average order of Rf (X).

1 Proposition 7.2.3. For α > 2 , we have

X 3 k+α+ 1 Z L(α + , f × f) X 2 1 α 2 2 k+α− 2 |Rf (t)| dt = 4 1 + O X . (7.5) 0 (2π) ζ(2α + 3) (k + α + 2 ) 123 Proof. By absolute convergence of the series (7.3) and asymptotics for Jk+α(z), we write the integral at left in (7.5) as

Z X 1 X a(n)a(m) 1 √ √ k+α− 2 −1 4 k α 1 t cos 4π nt − c cos 4π mt − c 1 + O t dt, 8π 2 + 2 + 4 n,m≥1 (mn) 0

π π 1 k+α− 2 in which c = 2 (k+α)+ 4 . The error term in the integrand contributes O(X ). As √ for the main term, we change variables u = t and expand each cosine in exponentials. Since Z X M+1 M iλu X M u e du = δ[λ=0] + O X , 0 M + 1

only the terms with m = n contribute to the main term. Evaluating these terms and bounding the terms with m 6= n gives the result.

1 Corollary 7.2.4. Choose α > 2 . Then:

α β k α 1 1. The inﬁmum of all β such that Rf (X) = O(X ) is 2 + 2 + 4 .

α β k α 1 2. The inﬁmum of all β such that Sf (X) = O(X ) is 2 + 2 + 4 .

7.3 Applications to (Non-Iterated) Partial Sums

The work of Hardy [Har17] and Jutila [Jut87] already demonstrates that higher Riesz

means may be used to control the unweighted sums Sk(X) and Sf (X). In this section we give a relation between the asymptotics of the various C´esaromeans and derive

upper and lower bounds for Sf (X) as a corollary.

Proposition 7.3.1. For integral m, let Sm(n) denote the mth C´esaro mean of the

m sequence {a(n)}. For each m, let ωm denote the inﬁmum of β such that S (X) =

β O(X ). Then ωm+1 + ωm−1 ≥ 2ωm for all integers m ≥ 0.

124 m ωm− Proof. Fix > 0. By the deﬁnition of ωm, we must have S (X) X for some sequence of X → ∞. Since Sm−1(X) = O(Xωm−1+), we have

m m ωm−1+ m |S (X + h)| |S (X)| − |h| · X S (X)

ωm−ωm−1−3 for all |h| ≤ Y := cX , where c > 0 is independent of X. We now estimate

X Sm+1(X + Y ) − Sm+1(X) = Sm(n) (7.6) n∈(X,Y ]

in two ways. Since ωm − ωm−1 ≤ 1 by the trivial summation bound, we see that the left-hand side of (7.6) is O(Sm+1(X)) = O(Xωm+1+). Conversely, the right-hand side

ωm− ωm−ωm−1−3 of (7.6) is X · X . Consistency forces ωm+1 + ≥ 2ωm − ωm−1 − 4, and the lemma follows by taking → 0.

0 We obtain two interesting results for Sf (X) = Sf (X) by combining the results of Corollary 7.2.4(2) with Proposition 7.3.1. In the case m = 1, we obtain upper bounds for Sf (X); in the case m = 0, we obtain lower bounds.

Corollary 7.3.2. Let f(z) be a cusp form of weight k and level 1. Then

k−1 + 3 + 1. Sf (X) = O X 2 8 .

k−1 + 1 − 2. Sf (X) = Ω X 2 4 .

1 k−1 + 3 + Proof. The ﬁrst claim follows from the bound Sf (X) = O(X 2 4 ) given in Corol- −1 lary 7.2.4(2) and the Deligne bound for Sf (n) = a(n). The second claim follows

1 2 directly from bounds for Sf (X) and Sf (X).

k−1 + 1 Remark 7.3.3. We recall that Sf (X) = O(X 3 3 ) following the work of [HI89] (with a further logarithmic savings due to [WX15]). Thus the upper bound given in Corollary 7.3.2(1) is somewhat weak. Conversely, the lower bound that appears in

125 Corollary 7.3.2(2) is conjecturally sharp (up to logarithmic factors, etc.).

Lower bounds in the generalized Gauss circle problem and cusp form analogy typically arise from on-average results or a synthesis of Voronoi formulas and Dirichlet’s approximation theorem. This thesis contains two new proofs of this lower bound in the cusp form analogy: one is here and the other is as a consequence of Corollary 3.5.2.

7.4 Dirichlet Series Attached to Riesz Means

A novel technique that may be used to study the Riesz means of sums of Fourier coeﬃcients of cusp forms involves the Dirichlet series

α α 2 α X Rf (n) α α X |Rf (n)| D(s, Rf ) := k−1 ,D(s, Rf × Rf ) := s+k−1 . s+ 2 n n≥1 n n≥1

Conveniently, the techniques of Chapter3 can be applied with very little modiﬁcation to study these series when α is integral. Each enjoys a meromorphic continuation to

α α C, moderate growth in vertical strips, and – in the case of D(s, Rf × Rf ) – a ﬁnite number of real poles to the right of an inﬁnite cluster of spectral poles.

On the other hand, it is unlikely that these series will shed light on any problems that the related, absolutely convergent Voronoi formulas cannot. For the sake of completeness, we conclude this chapter with a sketch the reduction of the series

α α D(s, Rf × Rf ) to familiar objects. For in the eternal words of a great mathematician,

“If you have a hammer, use it everywhere you can.”

— Benoit Mandelbrot

126 7.4.1 An Elementary Decomposition

α α As with D(s, Sf ×Sf ) in Chapter3, the ﬁrst step towards understanding D(s, Rf ×Rf ) is to decompose it algebraically into a collection of simpler functions. Adapting the proof of Proposition 3.1.2 yields the following.

Proposition 7.4.1. For integral α ≥ 1, we have

Z α α 1 Γ(z)Γ(s + k − 1 − z) D(s, Rf × Rf ) = Wf (s − z)ζ(z − 2α) dz 2πi (σ) Γ(s + k − 1) α 1 X αZ Γ(z)Γ(s + k − 1 − z) + Z (s − z, −j)ζ(z − 2α + j) dz, 2πi j f Γ(s + k − 1) j=1 (σ) in which σ > 2α + 1 and Re s σ.

α α Proof. Direct analogy with Proposition 3.1.2 shows that D(s, Rf ×Rf ) may be written

X X 1 X a(n)a(n)m2α+ a(n)a(n − `)mα(m + `)α (n + m)s+k−1 n≥1 m≥0 `≥1 X + a(n)a(n − `)mα(m + `)α . `≥1

Note that the terms with m = 0 vanish when α > 0. For the rest, we decouple (n + m)s+k−1 with the familiar Mellin–Barnes integral to obtain

1 Z X 1 X a(n)a(n)m2α + a(n)a(n − `)mα(m + `)α 2πi ns+k−1−zmz (σ) n,m≥1 `≥1 X Γ(z)Γ(s + k − 1 − z) + a(n)a(n − `)mα(m + `)α dz. Γ(s + k − 1) `≥1

Our result now follows by decoupling (m + `)α by the binomial theorem, collecting the m-sum into a zeta function, and collecting the n, `-sum into a Rankin–Selberg convolution and a shifted convolution.

127 α α 7.4.2 General Analytic Behavior of D(s, Rf × Rf )

α α Proposition 7.4.1 reduces the study of D(s, Rf ×Rf ) to that of a ﬁnite sum of integral transforms, which we study by shifting contours past the poles of ζ(z − 2α + j) and Γ(z) and extracting residues.

These residues involve the Rankin–Selberg transform L(s, f × f) as well as the

α α shifted convolutions Zf (s, −j). Thus D(s, Rf × Rf ) has a meromorphic continuation to C, built up from the meromorphic continuation of these terms.

α That being said, it is no easy task to describe the rightmost poles of D(s, Rf ×

α Rf ). The average order estimate from Corollary 7.2.4 (and a theorem due to Lan-

α α dau [Apo76, Theorem 11.13]) implies that the rightmost pole of D(s, Rf × Rf ) lies

3 at s = 2 + α, and Proposition 7.2.3 conﬁrms moreover that this pole has residue

L( 3 + α, f × f) 2 . (2π)4ζ(3 + 2α)

Far less clear is where this polar cancellation comes from, especially when α is large and the gap between the rightmost potential poles and the rightmost realized poles

α α of D(s, Rf × Rf ) grows in width.

1 1 7.4.3 Polar Analysis of D(s, Rf × Rf )

In this section, we give a sketch of the work required to classify the rightmost poles

α α α α of D(s, Rf × Rf ). Since the additional cancellation in D(s, Rf × Rf ) relative to that of D(s, Sf × Sf ) is increasingly technical (and decreasingly insightful) as α grows in size, we restrict ourselves to the case α = 1 for simplicity.

128 1 1 To further simplify matters, we limit our classiﬁcation to the poles of D(s, Rf ×Rf )

3 which lie in the half-plane Re s > 2 . This region includes the dominant pole(s) of 1 1 D(s, Rf × Rf ) and is chosen to omit the rightmost cluster of discrete spectral poles,

3 which in this case lies on Re s = 2 .

In the case α = 1, Proposition 7.4.1 implies that

Z 1 1 1 Γ(z)Γ(s + k − 1 − z) D(s, Rf × Rf ) = Wf (s − z)ζ(z − 2) dz 2πi (σ) Γ(s + k − 1) 1 Z Γ(z)Γ(s + k − 1 − z) + Zf (s − z, −j)ζ(z − 1) dz. 2πi (σ) Γ(s + k − 1)

1 Recall by Theorem 3.4.1 that Wf (s) is analytic in Re s > 2 . Similarly, we make 3 the claim that Zf (s, −1) is analytic in Re s > 2 . For the discrete spectral part of

Zf (s, −1), this follows at once from §3.3.1. For the continuous spectral part, it suﬃces

± to check for poles in R3/2(s, −1), which is described in (3.19). Thus contour shifting 1 1 3 reveals that the only poles of D(s, Rf ×Rf ) in Re s > 2 may be read from the residues

2Γ(s + k − 4) Γ(s + k − 3) W (s − 3) + Z (s − 2, −1) . (7.7) f Γ(s + k − 1) f Γ(s + k − 1)

Our treatment of potential poles in (7.7) requires several steps. In the ﬁrst, we

show that poles in the discrete spectral parts of Zf (s, 0) and Zf (s, −1) on the line

5 Re s = 2 cancel perfectly. We then classify the poles in the continuous spectral parts of Zf (s, 0) and Zf (s, −1).

Lemma 7.4.2. The total contribution of the two discrete spectral parts within (7.7)

5 is analytic on the line Re s = 2 .

129 Proof. The contribution of the discrete spectral parts of Zf (s, 0) and Zf (s, −1) within (7.7) may be written

(4π)k 2Γ(s + k − 4) X G(s − 3, it )L(s − 7 , µ )hV , µ i 2 Γ(s + k − 1) j 2 j f j j (4π)k Γ(s + k − 3) X + G(s − 2, it )L(s − 7 , µ )hV , µ i. 2 Γ(s + k − 1) j 2 j f j j

5 We compute the residue at s = 2 + itj to be

k (4π) L(−1 + it, µj)hVf , µji Γ(2it) 2Γ(−1 + 2it) 3 1 − 1 . (7.8) 2 Γ(k + 2 + it) Γ( 2 + it) Γ(− 2 + it)

Application of the Gauss duplication formula to Γ(2it) and Γ(−1 + 2it) shows that the parenthetical in (7.8) vanishes. Thus the discrete spectral contribution is in fact

5 5 analytic at s = 2 + itj, and the case s = 2 − itj is identical.

1 1 3 Our classiﬁcation of the poles of D(s, Rf × Rf ) in Re s > 2 via (7.7) requires 3 knowledge of the poles of Wf (s) in Re s > − 2 . Since Theorem 3.4.1 only documents 1 the poles of Wf (s) in Re s > − 2 , we describe a more detailed result now.

3 Lemma 7.4.3. The function Wf (s) is analytic in Re s > − 2 with the exception of 1 1 1 simple poles at s = 2 , s = − 2 and s = − 2 ± itj (for each type tj). We have

(k − 1 )L( 3 , f × f) Γ(k + 3 )L( 5 , f × f) Res W (s) = 2 2 and Res W (s) = 2 2 . f 2 f 4 3 s= 1 4π ζ(3) s=− 1 2 2 128π ζ(5)Γ(k − 2 )

Likewise, we require the meromorphic behavior of Zf (s, −1) in the right half-plane

1 ± ± Re s > − 2 , which amounts to inspecting R3/2(s, −1) and R1/2(s, −1) in the vertical 1 5 1 1 strips (− 2 , 2 ) and (− 2 , 2 ), respectively. We conclude the following.

130 1 Lemma 7.4.4. The function Zf (s, −1) is analytic in Re s > − 2 with the exception 3 1 1 of simple poles at s = 2 , s = 2 , and s = 2 ± itj (for each type tj). We have

L( 3 , f × f) 3Γ(k + 3 )L( 5 , f × f) Res Z (s, −1) = − 2 and Res Z (s, −1) = 2 2 . f 2 f 2 1 s= 3 2π ζ(3) s= 1 2 2 64π ζ(5)Γ(k − 2 )

1 1 3 Classiﬁcation of the poles of (7.7) (and thereby D(s, Rf × Rf )) in Re s > 2 now follows from Lemma 7.4.2, Lemma 7.4.3, and Lemma 7.4.4. For easy reference, these poles and their residues are presented in Table 7.1.

1 1 3 Table 7.1: Summary of Polar Data for D(s, Rf × Rf ) in the Half-Plane Re s > 2

pole location source of pole residue

1 3 7 2Γ(s+k−4) Γ(k+ 2 )L( 2 ,f×f) s = 2 Wf (s − 3) Γ(s+k−1) 2 5 2π ζ(3)Γ(k+ 2 ) 1 3 7 Γ(s+k−3) Γ(k+ 2 )L( 2 ,f×f) s = 2 Zf (s − 2, −1) Γ(s+k−1) − 2 5 2π ζ(3)Γ(k+ 2 ) 5 5 2Γ(s+k−4) L( 2 ,f×f) s = 2 Wf (s − 3) Γ(s+k−1) 64π4ζ(5) 5 5 Γ(s+k−3) 3L( 2 ,f×f) s = 2 Zf (s − 2, −1) Γ(s+k−1) 64π2ζ(5)

1 1 Theorem 7.4.5. The function D(s, Rf × Rf ) is meromorphic in C and analytic in 5 5 Re s > 2 . This function has a simple pole at s = 2 with residue

L( 5 , f × f) Res D(s, R1 × R1 ) = 2 5 f f 4 s= 2 (2π) ζ(5)

3 and is otherwise analytic in Re s > 2 .

5 Proof. Potential poles on the line Re s = 2 in the discrete spectral parts of Zf (s, 0)

and Zf (s, −1) cancel by Lemma 7.4.2. The remaining polar cancellation is visible in Table 7.1.

Remark 7.4.6. Theorem 7.4.5 can be used in conjunction with the Wiener–Ikehara

1 3/4 Theorem to give another proof that the average order of Rf (X) is X , and the lack of 131 secondary main terms is consistent with the size of the error term in Proposition 7.2.3.

Mostly, Theorem 7.4.5 should be considered as further proof that shifted convolution sums have a place in the study of partial sums of modular forms.

132 Appendix A

Explicit Bounds for D(s, P3 × P3)

This appendix presents a proof of explicit polynomial bounds for the growth of

1 D(s, P3 × P3) on the line Re s = σ, with σ ∈ (0, 2 ). This (extremely technical) result is used to produce explicit versions of Theorem 5.3.5 and Theorem 5.4.6 in the case k = 3. Our improvements appear as Theorem 5.3.7 and Theorem 5.4.9, respectively.

A.1 Explicit Bounds for W3(s)

We begin by producing bounds for W3(s) on vertical lines in the half-plane Re s > −3.

Since W3(s) may be collected as the standard Dirichlet series

2 X 2r3(n)S3(n) − r (n) W (s) = 3 , 3 ns+3 n≥1

1 + 3 the bounds r3(n) n 2 and S3(n) n 2 imply that W3(s) is absolutely convergent (and thus uniformly bounded on vertical lines) in the right half-plane Re s > 0.

133 Our analysis of W3(s) in the strip Re s ∈ (−3, 0) follows the decomposition of

W3(s) into diagonal, non-spectral, discrete, and continuous parts, and we treat each separately. Combining our bounds gives the following lemma.

Lemma A.1.1. The function W3(s) satisﬁes the explicit polynomial bounds

  5 (1 + |t|) 2 , −1 < σ   3 −σ+ W3(σ + it) σ (1 + |t|) 2 , −2 < σ < −1    − 1 −2σ+ (1 + |t|) 2 , −3 < σ < −2

for all > 0.

A.1.1 The Diagonal Part

The diagonal part of W3(s) is represented by an absolutely convergent series in

± 5 Re s > −1 and is completely cancelled by T1/2(s + 2 , 0) in the half-plane Re s < −2.

Bounds in the vertical strip −2 < Re s < −1 are obtained from the functional equation of L(s, f × f)/ζ(2s) and the Phragm´en–Lindel¨ofconvexity principle. Ex- plicitly, we have

 1, −1 < σ  2 5 k k  X r3(n) L(σ + + it, θ × θ )  2 −1−σ+ σ+it+3 = σ (1 + |t|) , −3 < σ < −1 n ζ(2σ + 5 + 2it)  n≥1   −4−2σ (1 + |t|) , σ < −3.

A.1.2 The Non-Spectral Part

In dimension k = 3, the non-spectral part of W3(s) appears as

4π5/2Γ(s + 1)ζ(s + 1)ζ(s + 3) 1 1 E (s) = 1 + − . 3 5 (2) 2s+3 s+2 Γ(s + 2 )ζ (3) 2 2 134 Bounds for E3(s) thus follow from Stirling’s approximation and estimates for ζ(s). Convexity estimates for ζ(s + 1) and ζ(s + 3) give

  − 3 (1 + |t|) 2 , 0 < σ    − 3 − σ + (1 + |t|) 2 2 , −1 < σ < 0 E3(σ + it) σ  −1−σ+ (1 + |t|) , −2 < σ < −1    −2− 3σ + (1 + |t|) 2 , −3 < σ < −2.

A.1.3 The Discrete Part

The discrete spectral part of W3(s) appears in the form

3 X Γ(s + 2 + itj)Γ(s + 2 − itj) (4π) 2 L(s + 2, µ )hV , µ i. (A.1) Γ(s + 5 )Γ(s + 3) j 3 j j 2

Let n0 > 0 be minimal such that ρj(n0) 6= 0. Our estimate for the discrete spectral term is derived by combining estimates for the gamma factors (ie. Stirling’s approximation), the uniform (in tj) convexity bound

  1, 1 < σ   −1 1 − σ + 1 − σ + ρj(n0) L(σ + it, µj) |t − tj| 2 2 |t + tj| 2 2 , 0 < σ < 1    1 −σ+ 1 −σ+ |t − tj| 2 |t + tj| 2 , σ < 0,

and an estimate for the size of |hV3, µjiρj(n0)|. Because this last term is hard to control directly, we work instead with the on-average bound

X 4 |hV3, µjiρj(n0)| T , (A.2)

|tj |

135 which follows by combining on-average bounds for hV3, µji given in [Kır15, Proposi-

tion 14] with on-average bounds for ρj(n0) given in [HHR16, (4.3)].

Stirling’s approximation reveals exponential decay with respect to |tj| in (A.1) in

the range |tj| > | Im s|. In the range |tj| < | Im s|, this exponential decay disappears and Abel summation gives

  5 (1 + |t|) 2 , −1 < σ   3 −σ+ (1 + |t|) 2 , −2 < σ < −1 (A.3)    − 1 −2σ+ (1 + |t|) 2 , σ < −2

for the discrete part on the line Re s = σ. A second application of Abel summation shows that these bounds also hold in the region |tj| > | Im s|.

Remark A.1.2. The bounds presented for the discrete spectrum in (A.3) are quite weak. It is then of no surprise that they represent the dominant terms in our bounds

for W3(s). Improved bounds in this section (and in particular, within (A.2)) are of considerable interest.

A.1.4 The Continuous Part

Our discussion of the continuous spectral part begins with the integral

3 5 1 +z (4π) 2 X Z G(s + , z)π 2 2 ζ (s + 5 , z) V ,E (·, 1 − z) dz. (A.4) 4πi Γ( 1 + z) a 2 3 a 2 a (0) 2

We bound the integrand in (A.4) by multiplying individual bounds for its factors. To this end, we remark that

1 1 3 3 hV3,Ea(·, − z)i Γ(1 − z) L( − z, θ |σa × θ |σa ) 3 2 2 2 + 1 1 · log(1 + |z|)(1 + |z|) Γ( 2 + z) Γ( 2 + z) ζ(1 + 2z) 136 by Stirling’s approximation, de la Vall´ee-Poussin’s estimate 1/ζ(1+it) log(1+|t|)|,

3 3 and convexity for L(z, θ |σa × θ |σa )/ζ(2z).

5 To bound the factor ζa(s, z), we note that ζa(σ + 2 + it, z) agrees with

ζ(σ + 2 + it − z)ζ(σ + 2 + it − z) ζ(1 + 2z) up to 2-factors and other elementary terms (as can be seen from (4.17)). By convexity for ζ(s) and de la Vall´ee-Poussin’s estimate, we conclude that

  1, −1 < σ   5 −σ − 1 + − σ − 1 + ζa(σ + 2 + it, z) log(1 + |z|) · |s − z| 2 2 |s + z| 2 2 , −2 < σ < −1    −σ− 3 + −σ− 3 + |s − z| 2 |s + z| 2 , σ < −2.

Accounting for G(s, z) as well, we conclude that (A.4) is

Z 3 +σ 3 +σ |s + z| 2 |s − z| 2 3 +3 −π max(|s|,|z|)+π|s| (1 + |z|) 2 e dz, if − 1 < σ 2σ+ 9 (0) |s| 2 Z 1+ σ + 1+ σ + |s + z| 2 |s − z| 2 3 +3 −π max(|s|,|z|)+π|s| (1 + |z|) 2 e dz, if − 2 < σ < −1 2σ+ 9 (0) |s| 2 Z |s + z| |s − z| 3 +3 −π max(|s|,|z|)+π|s| (1 + |z|) 2 e dz, if σ < −2. 2σ+ 9 (0) |s| 2

In each case, we divide the integral into ranges depending on the relative size of |s| and |z|. The contribution from |z| < |s| is dominant and produces the bound

  1+ (1 + |t|) , −1 < σ   (1 + |t|)−σ+, −2 < σ < −1    −2σ−2+ (1 + |t|) , σ < −2.

137 It remains to estimate the residual pairs on the lines for which they appear. The

± 5 residuals T3/2(s + 2 , 0) appear once σ < −1 and, following (4.22), take the form

3 1 3 3 2 s+1 + 5 − 5 (4π) Γ(s + 2)π Γ(−s) L(− 2 − s, θ × θ ) T 3 (s + 2 , 0) = −T 3 (s + 2 , 0) = 5 . 2 2 −s 2Γ(s + 2 )Γ(s + 3)(4π) ζ(−1 − 2s)

Stirling’s approximation and convexity bounds for L(s, θ3 ×θ3)/ζ(2s) give the bounds

  − 3 −σ (1 + |t|) 2 , −2 < σ < 0 ± 5  T 3 (s + 2 , 0) 2  − 7 −2σ (1 + |t|) 2 , σ < −2.

± 5 The second residual pair T1/2(s + 2 , 0) appears once σ < −2 but immediately cancels with the diagonal part, so we ignore it here. Further residuals do not appear in the half-plane Re s > −3.

A.2 Explicit Bounds for D(s, P3 × P3)

In this section, we use the bounds on W3(s) from Theorem A.1.1 to produce explicit

1 polynomial bounds for D(s, P3 × P3) on vertical lines in Re s > (0, 2 ). Our starting point is the decomposition

W (s − 1) D(s, S × S ) = ζ(s + 3) + 1 W (s) + 3 + 1 (s + 3)W (s + 1) (A.5) 3 3 2 3 s + 2 12 3 1 Z Γ(z)Γ(s + 3 − z) + W3(s − z)ζ(z) dz, (A.6) 2πi 5 Γ(s + 3) (− 2 )

which specializes a general formula given in §4.5.1.

The zeta function in (A.5) is bounded on vertical lines in the strip Re s ∈ (−2, −1), while growth of the simple terms involving W3(s) can be read from Theorem A.1.1.

For the integral in (A.6), we ﬁrst note that W3(s − z) lies in its half-plane of absolute

138 7 convergence. It follows by Stirling’s approximation that this integral is O(| Im s| 2 ). Taking worst-case bounds, we conclude the following:

7 Lemma A.2.1. Fix σ ∈ (−2, −1). Then D(σ + it, S3 × S3) (1 + |t|) 2 .

By combining Lemma A.2.1 with Proposition 4.1.1, we obtain polynomial bounds

for D(s, P3 × P3) in the region Re s ∈ (0, 1). To simplify the statement of our result,

1 we restrict to the strip Re s ∈ (0, 2 ).

1 7 2 Lemma A.2.2. Fix σ ∈ (0, 2 ). Then D(σ + it, P3 × P3) (1 + |t|) .

1 Proof. Fix σ ∈ (0, 2 ). In dimension k = 3, Proposition 4.1.1 reads

2 1 3 D(s, P3 × P3) = D(s − 2,S3 × S3) + v3ζ(s − 2) − 2v3ζ(s − 2 ) − 2v3L(s − 1, θ ) Z 1 v3 3 Γ(z)Γ(s − 2 − z) − L(s − 1 − z, θ )ζ(z) 1 dz, πi (2) Γ(s − 2 )

in which v3 = 4π/3. We bound D(σ + it, S3 × S3) using Lemma A.2.1 and bound the zeta functions above using functional equations and convexity bounds.

To bound the term L(s − 1, θ3), we apply the functional equation of L(s, θ3) given in (4.31) to write

3 Γ(2 − s) 3 k L(s − 1, θ ) 1 L( 2 − s, θ ). Γ(s − 2 )

1 3 3 Once σ < 2 , the L-function L( 2 − σ − it, θ ) enters into its half-plane of absolute convergence and is easily bounded. As for the Mellin–Barnes integral, we shift the

5 integral to (− 2 ) and extract residues corresponding to z = 1, 0, and −1. The sum of 7 −2σ these residues is (1 + |t|) 2 , as is the shifted integral, and we obtain our result by taking worst-case bounds.

139 Index

a, 52, 56 Deligne bound,7, 13, 16, 19, 125 a0, 71 δ[], the Kronecker delta, 33 Abel summation, 38, 136 diagonal term, 50, 67, 84 additive twist, 123 Dirichlet character, 14, 84, 116 αk,4, 44, 97 Dirichlet divisor problem,6, 116 discrete spectral contribution, 25 Bessel function discrete spectral part, 57, 82 J-Bessel function, 122, 123 divisor sum, twisted, 116 K-Bessel function, 16, 24, 52 βk, 109 e(z), 13 Borwein–Choi formula, 76, 85 E(z, s), 15 Ea(z, s), 52 C2, 90, 96 Ek(s), 71, 86, 134 0 C2, 90 Eisenstein series 0 C3 and C3, 96 holomorphic, 114 0 C4, 76, 96 of weight 1, 115 Ck, 79, 96 real analytic, 15, 52 Cahen–Mellin integral, 16, 39, 95, 98 Eisenstein series analogy, 118 C´esaromean, 120 Chandrasekharan–Narasimhan,7, 19, Fourier expansion 20, 38, 43 of Eisenstein series, 15, 52 concentrating integral, 103 of holomorphic Eisenstein series, congruence subgroup, 13 114 const, 81 of Maass forms, 24, 56 continuous spectral contribution, 25 functional equation continuous spectral part, 57, 63, 82 of Eisenstein series Ea(z, s), 65 cusp form, 13 of L(s, θk), 78 cusp form analogy, 18 of θ(z), 54

D(s, Pk × Pk), 46 G(s, z), 27, 57 D(s, Pk), 99, 108 Γ(n), 13 α α D(s, Rf × Rf ), 126 Γ0(n), 13 α D(s, Rf ), 126 γ, 71 D(s, Sk × Sk), 46 γk, 108, 111 D(s, Sf × Sf ), 19 Gauss circle problem,1 dµ(x), 17 generalized,3, 44 140 in the strict sense,3 Rankin–Selberg convolution, 17, 68 Gauss duplication formula, 35 ρj(n), 56 Riemann hypothesis, 30 Hecke operator, 13, 23 Riesz mean, 121

iterated partial sum, 120 Sα(n), 120 j-invariant, 14 Sf (n), 19, 124 Sk(n),3, 44 L(s, µj), 25 scaling matrix, 13, 52, 53 L(s, χ), 84, 117 scattering matrix, 65 L(s, f × f), 17, 21 second moment,3–5,7, 93, 107, 122, L(s, f), 16 123 L(s, µj), 56 discrete, 38, 93, 101 L(s, θk × θk), 67, 85 Selberg’s eigenvalue conjecture, 26, 62 L(s, θk), 48 Selberg’s spectral theorem, 45, 56 Laplace transform, 92, 96 σa, 52, 53 discrete, 42, 92, 95 σν(n), 16, 60 lattice point discrepancy,1 σν(n),5 linear fractional transformation, 12 slash operator, 14 square-root cancellation,2,7, 19, 123 Maass form, 24, 56 Mellin transform, 16, 102 T ±(s, w), 64, 66, 67, 70 Mellin–Barnes integral, 20, 49, 127 τ(n),7 M¨obiusfunction, 60 θ(z),6, 45, 53 modular discriminant,7 θSEC, 26 modular form, 12 uy(x) and Uy(s), 102, 108 non-spectral part, 57, 59, 82 Vf (z), 23 P (z, s), 16 h vk, 44, 47 P (n),3, 44 k Vk(z), 53, 80 Perron’s formula, 121 Voronoi formula,2, 118, 123, 126 Petersson inner product, 17, 23 phase change, 91, 113, 115 Wf (s), 21 ϕah(s), 52, 59 Wk(s), 50 Poincar´eseries, 16, 23 Watson’s triple product formula, 27 polynomial growth, 28, 40, 93, 106, Wiener–Ikehara, 38, 69, 93, 131 133 power-savings error, 39, 93, 105, 111 Zf (s, w), 21 Zk(s, w), 50 Rα(n), 121 Zagier regularization, 68, 85 R±(s, w), 32, 33, 40 ζ(s, w, z), 27 ∗ rk(n),6, 45, 116 ζ (s), 25 Ramanujan sum, 60 ζa(s, z), 63

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