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Large fixed order character sums

Introductory Workshop: Analytic Number Theory February 06, 2017 - February 10, 2017

February 09, 2017 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Youness Lamzouri (York University)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Abstract

For a non-principal Dirichlet character $\chi$ modulo $q$, the classical P\'olya-Vinogradov inequality asserts that $M(\chi):=\max_{x}|\sum_{n\leq x} \chi(n)|=O\left(\sqrt{q}\log q\right)$. This was improved to $\sqrt{q}\log\log q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Sasha Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the P\'olya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g\geq 3$ and conductor $q$, up to a power of $\log_4 q$ (where $\log_4$ is the fourth iterated logarithm).

Supplements
27974?type=thumb Lamzouri Notes 2.71 MB application/pdf Download
Video/Audio Files

Large Fixed Order Character Sums

H.264 Video 12-Lamzouri.mp4 695 MB video/mp4 rtsp://videos.msri.org/data/000/027/843/original/12-Lamzouri.mp4 Download
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