Large fixed order character sums
Introductory Workshop: Analytic Number Theory February 06, 2017  February 10, 2017
Location: SLMath: Eisenbud Auditorium
PolyaVinagradov inequality
character sums
Arithmetic functions
Large Fixed Order Character Sums
For a nonprincipal Dirichlet character $\chi$ modulo $q$, the classical P\'olyaVinogradov 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\'olyaVinogradov and MontgomeryVaughan 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).
Lamzouri Notes

Download 
Large Fixed Order Character Sums
H.264 Video 
12Lamzouri.mp4

Download 
Please report video problems to itsupport@slmath.org.
See more of our Streaming videos on our main VMath Videos page.