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High moments of theta functions and character sums
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Szabó, Barnabás (2024) High moments of theta functions and character sums. Mathematika, 70 (2). e12242. doi:10.1112/mtk.12242 ISSN 0025-5793.
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Mathematika - 2024 - Szabó - High moments of theta functions and character sums.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons: Attribution-Noncommercial 4.0. Download (343Kb) | Preview |
Official URL: http://doi.org/10.1112/mtk.12242
Abstract
Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet L‐functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related result ∫ 0 T | ζ ( 1 / 2 + i t ) | 2 k ≪ k T ( log T ) k 2 $\int _{0}^T|\zeta (1/2+it)|^{2k} \ll _k T(\log T)^{k^2}$ for all k ⩾ 0 $k\geqslant 0$ under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
SWORD Depositor: | Library Publications Router | ||||||||
Library of Congress Subject Headings (LCSH): | Number theory, Exponential sums, L-functions | ||||||||
Journal or Publication Title: | Mathematika | ||||||||
Publisher: | London Mathematical Society | ||||||||
ISSN: | 0025-5793 | ||||||||
Official Date: | April 2024 | ||||||||
Dates: |
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Volume: | 70 | ||||||||
Number: | 2 | ||||||||
Article Number: | e12242 | ||||||||
DOI: | 10.1112/mtk.12242 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Copyright Holders: | © 2024 The Authors. Mathematika is copyright © University College London and published by the London Mathematical Society on behalf of University College London. | ||||||||
Date of first compliant deposit: | 12 March 2024 | ||||||||
Date of first compliant Open Access: | 12 March 2024 | ||||||||
RIOXX Funder/Project Grant: |
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