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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.
