Paweł Nosal

On the distribution of very short character sums

In their paper concerning quadratic residues Davenport and Erd\H{o}s show that normalized sums of Legendre symbols $(\tfrac{n}{p})$ of suitable length $H(p) = p^{o(1)}$, with uniformly random starting point $X \in [0,...,p-1]$ obey the Central Limit Theorem, as the size of prime conductor goes to infinity.

Recently, Basak, Nath and Zaharescu proved that the CLT still holds, if we pick $X$ uniformly at random from $[0,...,(\log p)^A], A>1$ , set $H(p) = (\log p)^{o(1)}$ and take the limit along full density subset of primes.

In this talk, I will present a modification of their approach, inspired by the work of Harper on short character sums over moving intervals. This allows us to obtain the CLT of this type with $X$ uniformly random from $[0,...,g(p)]$ with practically arbitrary $g(p) \ll p^{\epsilon}$ for all $\epsilon >0$.