The distribution of short character sums

Abstract

Let $\chi$ be a non-real Dirichlet character modulo a prime $q$. In this paper we prove that the distribution of the short character sum $S_{\chi ,H}(x)={\sum }_{x<n\le x+H}\chi (n)$, as $x$ runs over the positive integers below $q$, converges to a two-dimensional Gaussian distribution on the complex plane, provided that $\log H=o(\log q)$ and $H\to \infty$ as $q\to \infty$. Furthermore, we use a method of Selberg to give an upper bound on the rate of convergence.

12 pages