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

Lamzouri, Youness; Mangerel, Alexander P.

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Authors

Youness Lamzouri



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 Mar 29, 2024
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
Related Public URLs https://ams.msp.org/articles/uploads/tran/accepted/180820-Lamzouri/180820-Lamzouri-v2.pdf

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