Joint IAS/Princeton University Number Theory Seminar

In the late 1950’s, Burgess developed a novel method for bounding short sums of a multiplicative Dirichlet character. This resulted in a subconvexity bound for Dirichlet $L$-functions on the critical line. In recent work, we have adapted the Burgess method to prove bounds for character sums in more general settings, and in particular via an unexpected application of the Vinogradov Mean Value Theorem. We will survey aspects of these methods, including joint works with D.R. Heath-Brown and J. Xu.