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

Abstract

The Gauss circle problem concerns the difference $P_2\left(n\right)$ 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 $P_2{\left(n\right)}^2$, and prove that this series has meromorphic continuation to $ℂ$. Using this series, we prove that the Laplace transform of $P_2{\left(n\right)}^2$ satisfies ${\int }_0^{\infty }{P}_2{\left(t\right)}^2{e}^{-t∕X}dt=C{X}^{3∕2}-X+O\left(X^{1∕2+𝜖}\right)$, 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 $r_2\left(n+h\right)r_2\left(n\right)$, where $h$ is fixed and $r_2\left(n\right)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for ${\sum }_{n\ge 1}{r}_2\left(n+h\right)r_2\left(n\right)e^{-n∕X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for ${\sum }_{n\le X}{r}_2\left(n+h\right)r_2\left(n\right)$.