Central limit theorems for random multiplicative functions

TL;DR

This paper investigates the distribution of sums of Steinhaus random multiplicative functions over various sets and shows that under certain conditions, these sums follow a central limit theorem, extending previous results on their non-Gaussian behavior.

Contribution

The paper introduces new conditions and sets for which a central limit theorem applies to sums of Steinhaus random multiplicative functions, broadening understanding of their probabilistic behavior.

Findings

Central limit theorem holds for sums over specific sets ${ m f A}$.

CLT applies to sums with irrational $ heta$ avoiding good Diophantine approximations.

Shows sums exhibit Gaussian distribution under new conditions.

Abstract

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in {\mathcal A}} f(n)$, where ${\mathcal A}$ is a subset of the integers in $[1,N]$, and produces several new examples of sets ${\mathcal A}$ where a central limit theorem can be established. We also consider more general sums such as $\sum_{n\le N} f(n) e^{2\pi i n\theta}$, where we show that a central limit theorem holds for any irrational $\theta$ that does not have extremely good Diophantine approximations.