Three conjectures about character sums

Granville, Andrew; Mangerel, Alexander P.

Abstract

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1, χ) and L(1+it, χ)) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

Citation

Granville, A., & Mangerel, A. P. (2023). Three conjectures about character sums. Mathematische Zeitschrift, 305(3), Article 49. https://doi.org/10.1007/s00209-023-03374-8

• Journal Article Type: Article
• Acceptance Date: Sep 15, 2023
• Online Publication Date: Oct 23, 2023
• Publication Date: 2023-11
• Deposit Date: Oct 25, 2023
• Publicly Available Date: Oct 25, 2023
• Journal: Mathematische Zeitschrift
• Print ISSN: 0025-5874
• Electronic ISSN: 1432-1823
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 305
• Issue: 3
• Article Number: 49
• DOI: https://doi.org/10.1007/s00209-023-03374-8
• Keywords: 11M06, Halász’s theorem, 11L40, Character sums, Pretentious number theory, Multiplicative functions, Dirichlet character
• Public URL: https://durham-repository.worktribe.com/output/1817315