Second Moments in the Generalized Gauss Circle Problem

Abstract

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 $k\ge 3$. We also obtain main terms and power-saving error terms for the sharp sums ${\sum }_{n\le X}{P}_k(n{)}^2$, 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.

Now 36 pages + 7 page appendix. To appear in Forum of Mathematics Sigma