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 $d\geq 2$, 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 $d$.