The distribution of large values of mixed character sums

TL;DR

This paper analyzes the distribution of large values of mixed character sums, providing precise estimates and bounds that support Montgomery's conjecture, with notable differences observed between even and odd character orders.

Contribution

It offers new bounds and distribution estimates for mixed character sums, extending previous results and supporting conjectures about Fekete polynomial maxima.

Findings

Distribution function exhibits double-exponential decay.

Significant differences between even and odd order characters.

Improved bounds for the maximum of character sums.

Abstract

In this paper, we investigate the distribution of values of the complete exponential sum $S_{p,\chi}(\theta)=\sum_{n=1}^p \chi(n)e(n\theta)$, where $p$ is a large prime, $\chi$ is a Dirichlet character (mod $p$) of order $d\geq 2$, and $\theta$ varies over certain subsets of $[0,1]$. When $d=2$, these sums correspond to the values of the Fekete polynomial associated with $p$ on the unit circle. Our first result gives precise estimates for the tail of the distribution of $|S_{p,\chi}(\theta)|$ in a large uniform range, when $\theta$ varies over the set $\{(k+1/2)/p\}_{1\leq k\leq p}$. This improves upon a result of Conrey, Granville, Poonen, and Soundararajan. We also consider the distribution of the maximum of $|S_{p,\chi}(\theta)|$ for $\theta\in I_k=[k/p,(k+1)/p]$, and obtain upper and lower bounds for the distribution of large values of this maximum, valid in a uniform range that is nearly optimal: we make this precise in the paper. Our results provide strong support for a conjecture of Montgomery on the maximum of Fekete polynomials on the unit circle. In particular, we show that the distribution function exhibits double-exponential decay, with a surprising difference in behavior between the cases of even and odd order $d$.