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SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

Published online by Cambridge University Press:  19 December 2018

THOMAS A. HULSE
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02467, USA; thomas.hulse@bc.edu
CHAN IEONG KUAN
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong Province, 519082, China; kuanchi3@mail.sysu.edu.cn
DAVID LOWRY-DUDA
Affiliation:
Warwick Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK; d.lowry@warwick.ac.uk
ALEXANDER WALKER
Affiliation:
Department of Mathematics, Rutgers University Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA; alexander.walker@rutgers.edu

Abstract

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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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

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