Almost sure lower bounds for a model problem for multiplicative chaos in number theory

TL;DR

This paper establishes almost sure lower bounds for a simplified model of multiplicative chaos in number theory, extending Harper's results to a function field analogue and making the proof more accessible.

Contribution

It provides the first almost sure lower bounds in a simplified multiplicative chaos model, answering a question from prior work and adapting Harper's approach to a function field setting.

Findings

Proved almost sure lower bounds for the model problem.

Extended Harper's results to a function field analogue.

Made the proof more self-contained and accessible.

Abstract

The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self-contained, and perhaps somewhat more accessible.