  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.
