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.