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

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Sums of Fourier Coefficients of Modular Forms and the Gauss Circle Problem 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 jθ (z)j 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)...................
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