A lower bound on high moments of character sums

TL;DR

This paper establishes a near-optimal lower bound for high moments of Dirichlet character sums and theta functions, advancing understanding of their extremal behavior under the Generalised Riemann Hypothesis.

Contribution

It provides the first sharp lower bounds for high moments of character sums and theta functions, matching conjectured optimal bounds up to constants.

Findings

Lower bound on high moments of Dirichlet character sums.

Sharp lower bounds on moments of theta functions.

Results are optimal up to a constant factor under GRH.

Abstract

For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x} \chi(n)\Big|^{2k}, \end{equation*} where $1\leq x\leq q$, and $\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method.