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

The Gauss circle problem concerns the difference P-2(n) between the area of a circle of radius root 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 C. Using this series, we prove that the Laplace transform of P-2(n)(2) satisfies integral(infinity)(0) P-2(t)(2)e(-t/X) dt = CX3/2 - X + O(X1/2+epsilon), which gives a power-savings improvement to a previous result of Ivic (1996). 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 Sigma(n >= 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 Sigma(n <= X) r(2)(n + h)r(2)(n).