The distribution of short character sums

Abstract

AbstractLet χ be a non-real Dirichlet character modulo a prime q. In this paper we prove that the distribution of the short character sum Sχ,H(x) = ∑x<n≤x+H χ(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 → ∞ as q → ∞. Furthermore, we use an idea of Selberg to establish an upper bound on the rate of convergence.