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2021 The Laplace transform of the second moment in the Gauss circle problem
Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda, Alexander Walker
Algebra Number Theory 15(1): 1-27 (2021). DOI: 10.2140/ant.2021.15.1

Abstract

The Gauss circle problem concerns the difference P2(n) between the area of a circle of radius n and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients P2(n)2, and prove that this series has meromorphic continuation to . Using this series, we prove that the Laplace transform of P2(n)2 satisfies 0P2(t)2etXdt=CX32X+O(X12+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 r2(n+h)r2(n), where h is fixed and r2(n) denotes the number of representations of n as a sum of two squares. We use this Dirichlet series to prove asymptotics for n1r2(n+h)r2(n)enX, and to provide an additional evaluation of the leading coefficient in the asymptotic for nXr2(n+h)r2(n).

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Thomas A. Hulse. Chan Ieong Kuan. David Lowry-Duda. Alexander Walker. "The Laplace transform of the second moment in the Gauss circle problem." Algebra Number Theory 15 (1) 1 - 27, 2021. https://doi.org/10.2140/ant.2021.15.1

Information

Received: 30 June 2017; Revised: 17 June 2020; Accepted: 21 July 2020; Published: 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.2140/ant.2021.15.1

Subjects:
Primary: 11F30
Secondary: 11E45, 11F27, 11F37

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 1 • 2021
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