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The distribution of
large quadratic character sums and applications
Youness Lamzouri |
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Vol. 18 (2024), No. 11, 2091–2131
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Abstract |
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We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants . In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to , and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to modulo . Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols. |
Keywords
character sums, Dirichlet L-functions, Pólya–Vinogradov
inequality, quadratic large sieve, Kronecker symbol,
Legendre symbol, positivity of partial sums
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Mathematical Subject Classification
Primary: 11L40, 11N64
Secondary: 11K65
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Milestones
Received: 2 March 2023
Revised: 12 September 2023
Accepted: 27 November 2023
Published: 18 October 2024
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Authors |
| Youness Lamzouri | |
| Institut Élie Cartan de
Lorraine Université de Lorraine, CNRS Nancy France |
IRL3457 CRM-CNRS Centre de Recherches Mathématiques Université de Montréal Montréal, QC Canada |
| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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