Joint IAS/Princeton University Number Theory Seminar

Sums of Dirichlet characters $\sum_{n \leq x} \chi(n)$ (where $\chi$ 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 $\frac{1}{r-1} \sum_{\chi \ \text{mod}\ r} |\sum_{n \leq x} \chi(n)|^{2q}$, 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 $0 \leq q \leq 1$. I will focus mainly on the number theoretic issues arising.