The Gauss circle problem concerns the difference between the area of a circle of radius and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients , and prove that this series has meromorphic continuation to . Using this series, we prove that the Laplace transform of satisfies , 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 , where is fixed and denotes the number of representations of as a sum of two squares. We use this Dirichlet series to prove asymptotics for , and to provide an additional evaluation of the leading coefficient in the asymptotic for .
"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