Average first-passage times for character sums

TL;DR

This paper studies the behavior of the first-passage times of character sums over primes, establishing an asymptotic formula for their sum as primes grow large.

Contribution

It introduces the concept of first-passage times for character sums and proves their asymptotic distribution over primes.

Findings

Asymptotic formula for sum of first-passage times over primes

Existence of a constant c_ε relating to the distribution

Behavior of character sums crossing a linear threshold

Abstract

Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}. $$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$