Event
Youness Lamzouri, York University
Thursday, March 16, 2017 11:00to12:30
Room 5448, Pavillon Andr茅-Aisenstadt, 2920, Chemin de la tour, 5th floor, Montreal, QC, H3T 1J4, CA
Large Character Sums
For a non-principal Dirichlet character聽聽modulo聽, the classical Polya-Vinogradov inequality asserts that聽. This was improved to聽聽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聽, 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聽, when聽聽has odd order聽聽and conductor聽, up to a power of聽聽(where聽聽is the fourth iterated logarithm).