The distribution of quadratic character sums and applications

In this talk, I will present recent results on the distribution of the maximum of quadratic character sums, as well as some applications. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of non-principal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve.

We shall also describe two applications of our results. The first concerns the positivity of sums of the Legendre symbol, a question that was considered by Montgomery. The second, joint with Ayesha Hussain, investigates the distribution of "Legendre paths", which are character paths formed with the Legendre symbol.