Large values of short character sums

TL;DR

This paper proves that for any positive A, there are infinitely many primes p where the sum of the Legendre symbol over an interval of length (ln p)^A can be arbitrarily large.

Contribution

It establishes the existence of infinitely many primes with large Legendre symbol sums over specific short intervals, extending understanding of character sums.

Findings

Existence of infinitely many primes with large character sums

Character sums can be arbitrarily large over short intervals

Quantitative bounds on sum sizes for primes

Abstract

In this paper, we prove that for any $A>0$ there exist infinitely many primes $p$ for which sums of the Legendre symbol modulo $p$ over an interval of length $(\ln p)^A$ can take large values.