On the distribution of very short character sums

2025/12/02 by Nosal, Paweł
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.02915

Abstract

We establish a central limit theorem of (1/√(h_p))∑_{X< n ≤ X+hp}(\tfracnp) 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))→ 0, h_p → ∞ as p → ∞. 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.

Citations

• A note on character sums over short moving intervals
• Burgess bounds for short character sums evaluated at forms II: the mixed case
• Large character sums: Burgess's theorem and zeros of L-functions
• Conditional bounds for the least quadratic non-residue and related\n problems
• The distribution of short character sums
• Large character sums: Pretentious characters and the Polya-Vinogradov Theorem
• The Difference Between Consecutive Primes, II
• Large character sums

Related

• The distribution of short character sums
• A note on character sums over short moving intervals
• The Limiting Distribution of Character Sums
• The typical size of character and zeta sums is o(√(x))
• Central Limit Theorems for series of Dirichlet characters