Youness Lamzouri: Large character sums

Abstract : For a non-principal Dirichlet character χ modulo q, the classical Pólya-Vinogradov inequality asserts that
M(χ):=maxx|∑n≤xχ(n)|=O(q‾√log q).
This was improved to 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(χ), extending and refining Paley's construction. The second part, joint with Alexander 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 Pólya-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(χ), when χ has odd order g≥3 and conductor q, up to a power of log4q (where log4 is the fourth iterated logarithm).

Recording during the thematic meeting : "Prime Numbers and Automatic Sequences : Determinism and Randomness" the May 24, 2017 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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