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 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.
Walker, Alexander Walker,
"Sums of Fourier Coefficients of Modular Forms and the Gauss Circle Problem"
(2018).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/pgfp-kg69
