Sign changes of the partial sums of a random multiplicative function

TL;DR

This paper proves that the partial sums of a Rademacher random multiplicative function change sign infinitely often with probability one as the input grows large.

Contribution

It offers a simple proof establishing the almost sure infinite sign changes of partial sums for Rademacher random multiplicative functions.

Findings

Partial sums change sign infinitely often almost surely.

The proof is notably simple compared to previous methods.

The result confirms a long-standing conjecture in probabilistic number theory.

Abstract

We provide a simple proof that the partial sums $\sum_{n\leq x}f(n)$ of a Rademacher random multiplicative function $f$ change sign infinitely often as $x\to\infty$, almost surely.