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 $k\geqslant 3$ . We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$ , 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-3 Gauss circle problem.