# Second Moments in the Generalized Gauss Circle Problem

## TL;DR

This paper advances understanding of the lattice point discrepancy in the generalized Gauss circle problem by deriving asymptotics with power-saving error terms for sums involving squared discrepancies, including the first such result in three dimensions.

## Contribution

It provides new asymptotic formulas with power-saving error terms for sums of squared discrepancies in the generalized Gauss circle problem, notably achieving the first power-saving error in three dimensions.

## Key findings

- Improved asymptotics for smoothed sums of squared discrepancies.
- Power-saving error terms for sharp sums in dimensions ≥ 3.
- First power-saving mean square error in 3D 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 $k \geq 3$. 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.