Lower bounds for low moments of character sums

I will discuss the problem of bounding moments (that is, power averages) of sums $\sum_{n \leq x} \chi(n)$, where $\chi$ varies over all non-principal Dirichlet characters mod $r$. More specifically, I will be interested in the "low moments" (up to the second moment). One expects these to be well modelled by the corresponding moments of Steinhaus random multiplicative functions, and in previous work I proved upper bounds that confirm this. I will report on work in progress that should (hopefully) obtain matching lower bounds, and also describe a potential application to non-vanishing.