Large odd order character sums and improvements of the P\'olya-Vinogradov inequality

Lamzouri, Youness; Mangerel, Alexander P.

Abstract

For a primitive Dirichlet character modulo q, we dene M() = maxt j P nt (n)j. In this paper, we study this quantity for characters of a xed odd order g 3. Our main result provides a further improvement of the classical Polya-Vinogradov inequality in this case. More specically, we show that for any such character we have M() " p q(log q)1􀀀g (log log q)􀀀1=4+"; where g := 1􀀀g sin(=g). This improves upon the works of Granville and Soundarara- jan and of Goldmakher. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that M() p q (log2 q)1􀀀g (log3 q)􀀀1 4 (log4 q)O(1) ; where logj is the j-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of log4 q). One of the key ingredients in the proof of the upper bounds is a new Halasz-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.

Citation

Lamzouri, Y., & Mangerel, A. P. (2022). Large odd order character sums and improvements of the P\'olya-Vinogradov inequality. Transactions of the American Mathematical Society, 375, 3759-3793. https://doi.org/10.1090/tran/8607

• Journal Article Type: Article
• Acceptance Date: Nov 26, 2019
• Online Publication Date: Mar 4, 2022
• Publication Date: 2022
• Deposit Date: Oct 20, 2021
• Publicly Available Date: Aug 10, 2022
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 375
• Pages: 3759-3793
• DOI: https://doi.org/10.1090/tran/8607
• Public URL: https://durham-repository.worktribe.com/output/1226046
• Related Public URLs: https://ams.msp.org/articles/uploads/tran/accepted/180820-Lamzouri/180820-Lamzouri-v2.pdf