Second Moments in the Generalized Gauss Circle Problem
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 poin...
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| description | 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. |
| doi_str_mv | 10.48550/arxiv.1703.10347 |
| format | Article |
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| subjects | Data envelopment analysis Dirichlet problem Energy conservation Errors Laplace transforms Mathematics - Number Theory Sums |
| title | Second Moments in the Generalized Gauss Circle Problem |
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