The Laplace Transform of the Second Moment in the Gauss Circle Problem

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 $\mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n{)}^2$ satisfies ${\int }_0^{\infty }{P}_2(t{)}^2{e}^{-t/X}dt=C{X}^{3/2}-X+O(X^{1/2+\epsilon })$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. 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 ${\sum }_{n\ge 1}{r}_2(n+h)r_2(n)e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for ${\sum }_{n\le X}{r}_2(n+h)r_2(n)$.

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