Sums of Dirichlet characters ∑n≤xχ(n) (where χ is a character modulo some prime r, say) are one of the best-studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments 1/(r−1) ∑χ mod r|∑n≤xχ(n)|2q, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when 0≤q≤1. I will focus mainly on the number-theoretic issues arising.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
