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Second moments in the generalized Gauss circle problem
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Hulse, Thomas, Chan, Ieong Kuan, Lowry-Duda, David and Walker, Alexander (2018) Second moments in the generalized Gauss circle problem. Forum of Mathematics: Sigma, 6 . e24. doi:10.1017/fms.2018.26 ISSN 2050-5094.
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Official URL: https://doi.org/10.1017/fms.2018.26
Abstract
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 √ 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 ∑ P k ( n ) 2 e − n/X and the Laplace transform ∫ ∞ 0 P k ( t ) 2 e − t/X dt , in dimensions k ≥ 3. We also obtain main terms and power-saving error terms for the sharp sums ∑ n ≤ X P k ( n ) 2 , along with similar results for the sharp integral ∫ X 0 P 3 ( t ) 2 dt. This includes producing the first power-saving error term in mean square for the dimension-three Gauss circle problem.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Dirichlet series, Geometry -- Famous problems | ||||||||||||
Journal or Publication Title: | Forum of Mathematics: Sigma | ||||||||||||
Publisher: | Cambridge University Press (CUP) | ||||||||||||
ISSN: | 2050-5094 | ||||||||||||
Official Date: | 19 December 2018 | ||||||||||||
Dates: |
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Volume: | 6 | ||||||||||||
Article Number: | e24 | ||||||||||||
DOI: | 10.1017/fms.2018.26 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
Date of first compliant deposit: | 15 November 2018 | ||||||||||||
Date of first compliant Open Access: | 15 November 2018 | ||||||||||||
RIOXX Funder/Project Grant: |
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