On the distribution of very short character sums

We establish a central limit theorem of $(1\mathrm{/}\sqrt{h_p}){\sum }_{X<n\le X+h_p}\left(\frac{n}{p}\right)$ 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 $(\log h_p)\mathrm{/}(\log g(p))\to 0,h_p\to \mathrm{\infty }$ as $p\to \mathrm{\infty }$. 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 Erdős 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.