Random multiplicative functions and typical size of character in short intervals

TL;DR

This paper investigates the behavior of sums of random multiplicative functions in short intervals, identifying when they outperform square-root cancellation, and extends results related to character sums by modeling characters as random multiplicative functions.

Contribution

It provides a sharp bound for character sums in short intervals by modeling characters with random multiplicative functions, extending Harper's results.

Findings

Square-root cancellation diminishes when log(x/y) is around sqrt(log log x)

Established a sharp bound for character sums in short intervals

Extended Harper's results on character sums using probabilistic models

Abstract

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by $\sum_{x<n \leqslant x+y} f(n)$, exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio $\log\big(\frac{x}{y}\big)$ is around $\sqrt{\log\log x}$. By modeling characters by random multiplicative functions, we give a sharp bound of $\frac{1}{r-1}\sum_{\chi \!\!\!\mod r} \big|\sum_{x<n\leqslant x+y}\chi(n)\big|$, where $r$ is a large prime and $x+y\leqslant r $. This extends the result of Harper \cite{Harper_charac}.