On the distribution of very short character sums
Paweł Nosal
TL;DR
The paper proves a central limit theorem for very short Dirichlet character sums with a moving starting point, showing that, for almost all primes p and starting points X chosen uniformly from [g(p)], the normalized sum converges to a standard normal under mild growth conditions on g and h_p. The authors improve prior results by using a Selberg sieve framework and an extended random multiplicative function to control variances, enabling shorter averaging intervals (η up to 0.01) and extending the result to all primes via Mauduit–Sárközy-type bounds. Two main technical pillars underpin the work: (i) universal bounds on Diophantine representations from Evertse–Silverman and (ii) sharp prime-gap results from Baker–Harman–Pintz, supplemented by a Davenport–Erdős–style method of moments. The findings advance understanding of the distribution of short character sums and highlight the current barriers to extending these results to even smaller starting-interval sizes or to all-primes regimes without additional breakthroughs in incomplete-sum bounds. Practical impact lies in refined probabilistic models for character sums and potential applications to problems such as locating small quadratic non-residues and related distributional questions in analytic number theory.
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 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.