Better than square-root cancellation for random multiplicative functions

TL;DR

This paper explores the conditions under which the sum of a random multiplicative function over rough numbers exhibits better than square-root cancellation, identifying a specific threshold related to the roughness parameter.

Contribution

It establishes the precise threshold for the disappearance of better than square-root cancellation in sums involving random multiplicative functions over rough numbers.

Findings

Identifies the threshold $oxed{ ext{log} ext{log} R ot hickapprox ( ext{log} ext{log} x)^{1/2}}$ for cancellation phenomena.

Shows the phenomenon's presence depends on the roughness parameter R.

Provides insight into the behavior of sums involving random multiplicative functions over specific number sets.

Abstract

We investigate when the better than square-root cancellation phenomenon exists for $\sum_{n\le N}a(n)f(n)$, where $a(n)\in \mathbb{C}$ and $f(n)$ is a random multiplicative function. We focus on the case where $a(n)$ is the indicator function of $R$ rough numbers. We prove that $\log \log R \asymp (\log \log x)^{\frac{1}{2}}$ is the threshold for the better than square-root cancellation phenomenon to disappear.