Published online by Cambridge University Press: 19 December 2018
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.
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