The Laplace transform of the second moment in the Gauss circle problem

Hulse, Thomas A.; Kuan, Chan Ieong; Lowry-Duda, David; Walker, Alexander

doi:10.2140/ant.2021.15.1

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The Laplace transform of the second moment in the Gauss circle problem

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda and Alexander Walker

Vol. 15 (2021), No. 1, 1–27

DOI: 10.2140/ant.2021.15.1

Abstract

The Gauss circle problem concerns the difference $P_{2} (n)$ between the area of a circle of radius $\sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_{2} {(n)}^{2}$ , and prove that this series has meromorphic continuation to $ℂ$ . Using this series, we prove that the Laplace transform of $P_{2} {(n)}^{2}$ satisfies $\int_{0}^{\infty} P_{2} {(t)}^{2} e^{- t ∕ X} d t = C X^{3 ∕ 2} - X + O (X^{1 ∕ 2 + 𝜖})$ , which gives a power-savings improvement to a previous result of Ivić (1996).

Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_{2} (n + h) r_{2} (n)$ , where $h$ is fixed and $r_{2} (n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $\sum_{n \geq 1} r_{2} (n + h) r_{2} (n) e^{- n ∕ X}$ , and to provide an additional evaluation of the leading coefficient in the asymptotic for $\sum_{n \leq X} r_{2} (n + h) r_{2} (n)$ .

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Keywords

Gauss circle problem, modular forms, automorphic forms, multiple Dirichlet series

Mathematical Subject Classification 2010

Primary: 11F30

Secondary: 11E45, 11F27, 11F37

Milestones

Received: 30 June 2017

Revised: 17 June 2020

Accepted: 21 July 2020

Published: 1 March 2021

Authors

Thomas A. Hulse
Department of Mathematics Boston College Chestnut Hill, MA United States

Chan Ieong Kuan
School of Mathematics Sun Yat-Sen University Zhuhai China

David Lowry-Duda
ICERM and Brown University Providence, RI United States

Alexander Walker
Department of Mathematics Rutgers University Piscataway, NJ United States

Vol. 15, No. 1, 2021