The distribution of short character sums

TL;DR

This paper proves that short character sums for non-real Dirichlet characters distribute as a two-dimensional Gaussian in the complex plane under certain conditions, with an analysis of the convergence rate.

Contribution

It establishes the Gaussian distribution of short character sums and provides a method to bound the convergence rate, extending understanding of character sum behavior.

Findings

Distribution converges to 2D Gaussian as q→∞

Conditions: log H = o(log q) and H→∞

Provides bounds on convergence rate

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 $S_{\chi,H}(x)=\sum_{x< n\leq x+H} \chi(n)$, 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.