Better than squareroot cancellation in number theory

TL;DR

This paper surveys the phenomenon of better than square root cancellation in number theory, especially in multiplicative character sums, highlighting its connection with multiplicative chaos and potential applications.

Contribution

It provides a concise overview of better than square root cancellation, emphasizing number theoretic insights and the link to multiplicative chaos.

Findings

Discussion of averages of multiplicative character sums

Connection between cancellation phenomena and multiplicative chaos

Potential applications in number theory

Abstract

We give a short survey of the phenomenon of better than squareroot cancellation, specifically as it applies to averages of multiplicative character sums (such as $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$) thanks to their connection with so-called multiplicative chaos. We focus on the number theoretic aspects of the arguments, and also touch on some possible applications.