# [Liste-CICMA] Analytic Number Theory Seminar

Steve Lester sjlester at gmail.com
Mon Mar 13 11:09:45 EDT 2017

Dear All,

This Thursday, March 16 Youness Lamzouri from the York University  will be
speaking in the Analytic Number Theory Seminar from 11:00-12:30 in André

Title: Large Character Sums

Abstract: For a non-principal Dirichlet character [image: \chi] modulo [image:
q], the classical Polya-Vinogradov inequality asserts that [image:
M(\chi)=\max_x |\sum_{n \le x} \chi(n)|=O(\sqrt{q} \log q)]. This was
improved to [image: \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
characters sums. In the first part, we discuss even order characters, in
which case we obtain optimal omega results for [image: 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 Polya-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 [image:
M(\chi)], when [image: \chi] has odd order [image: g\ge 3] and
conductor [image:
q], up to a power of [image: \log_4 q] (where [image: \log_4] is the fourth
iterated logarithm).

Kind regards,
Steve Lester
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