Vol. 15, No. 1, 2021

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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

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 = CX32 X + 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).

Gauss circle problem, modular forms, automorphic forms, multiple Dirichlet series
Mathematical Subject Classification 2010
Primary: 11F30
Secondary: 11E45, 11F27, 11F37
Received: 30 June 2017
Revised: 17 June 2020
Accepted: 21 July 2020
Published: 1 March 2021
Thomas A. Hulse
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Chan Ieong Kuan
School of Mathematics
Sun Yat-Sen University
David Lowry-Duda
ICERM and Brown University
Providence, RI
United States
Alexander Walker
Department of Mathematics
Rutgers University
Piscataway, NJ
United States