Bounds for moments of quadratic character sums and theta functions

Abstract

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a multivariate tauberian theorem. As a consequence, we prove unconditional lower bounds for all even integer moments of quadratic character sums in a wide range of parameters. Moreover, assuming the Generalised Riemann Hypothesis (GRH), we prove a sharp upper bound on moments of character sums of arbitrary length. In a similar fashion, we obtain unconditional lower bounds on moments of quadratic theta functions and matching conditional upper bounds under GRH. In the case of the second moment of theta functions, we prove an optimal upper bound unconditionally improving the previous results of Louboutin and the first named author.