The distribution of large values of mixed character sums

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 $d\ge 2$, 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 $\mathrm{\mid }{S}_{p,\chi }(\theta )\mathrm{\mid }$ in a large uniform range, when $\theta$ varies over the set $\{(k+1\mathrm{/}2)\mathrm{/}p{\}}_{1\le k\le p}$. This improves upon a result of Conrey, Granville, Poonen, and Soundararajan. We also consider the distribution of the maximum of $\mathrm{\mid }{S}_{p,\chi }(\theta )\mathrm{\mid }$ for $\theta \in I_k=[k\mathrm{/}p,(k+1)\mathrm{/}p]$, 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 $d$.