#### 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
##### 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 ${\int }_{0}^{\infty }{P}_{2}{\left(t\right)}^{2}{e}^{-t∕X}\phantom{\rule{0.3em}{0ex}}dt=C{X}^{3∕2}-X+O\left({X}^{1∕2+𝜖}\right)$, 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 ${\sum }_{n\le X}{r}_{2}\left(n+h\right){r}_{2}\left(n\right)$.

##### Keywords
Gauss circle problem, modular forms, automorphic forms, multiple Dirichlet series
##### Mathematical Subject Classification 2010
Primary: 11F30
Secondary: 11E45, 11F27, 11F37