On the distribution of very short character sums

TL;DR

This paper proves a central limit theorem for very short character sums over primes, extending previous results to more general and shorter intervals, using advanced sieve and character sum techniques.

Contribution

It introduces a broader central limit theorem for short character sums over primes with minimal growth conditions on the interval length.

Findings

Establishes a CLT for character sums over almost all primes in very short intervals.

Extends the CLT to intervals with minimal growth conditions on length.

Uses advanced sieve methods and bounds on character sums to achieve results.

Abstract

We establish a central limit theorem of $(1/\sqrt{h_p})\sum_{X< n \leq X+h_p}\big(\tfrac{n}{p}\big)$ for almost all the primes $p$, with $X$ uniformly random in $[g(p)]$, $g(p)$ an arbitrary divergent function growing slower than any power of $p$, provided $(\log h_p)/(\log g(p))\rightarrow 0, \, h_p \rightarrow \infty$ as $p \rightarrow \infty$. This improves the recent results of Basak, Nath and Zaharescu, who established this for $g(p) = (\log p)^A, A>1$. We also use the best currently available tools to expand the original central limit theorem of Davenport and Erd\H{o}s for all the primes to a shorter interval of starting points. In this paper we exploit a Selberg's sieve argument, recently used by Harper, an intersection result due to Evertse and Silverman and some consequences of the Weil bound on general character sums.