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arXiv · 2409.13436

A lower bound on high moments of character sums

Barnabás Szabó
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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.

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Barnabás Szabó. 2024-09-20. A lower bound on high moments of character sums. https://arxiv.org/abs/2409.13436

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