Second Moments in the Generalized Gauss Circle Problem

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to \(P_k(n)^2\), where \(P_k(n)\) is the discrepancy between the volume of the \(k\)-dimensional sphere of radius \(\sqrt{n}\) and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including \(\sum P_k(n)^2 e^{-n/X}\) and the Laplace transform \(\int_0^\infty P_k(t)^2 e^{-t/X}dt\), in dimensions \(k \geq 3\). We also obtain main terms and power-saving error terms for the sharp sums \(\sum_{n \leq X} P_k(n)^2\), along with similar results for the sharp integral \(\int_0^X P_3(t)^2 dt\). This includes producing the first power-saving error term in mean square for the dimension-three Gauss circle problem.