A model problem for multiplicative chaos in number theory

TL;DR

This paper explores a simplified model related to multiplicative chaos in number theory, inspired by Harper's work on partial sums of random multiplicative functions, aiming to provide an accessible entry point into this complex area.

Contribution

The paper develops a simplified, function field version of Helson's conjecture, extending Harper's ideas to make the concepts more approachable for newcomers.

Findings

Simplified proof of a model problem related to multiplicative chaos

Connection established between number theory and probability theory concepts

Potential groundwork for future research in multiplicative chaos and number theory

Abstract

Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.