Almost sure bounds for a weighted Steinhaus random multiplicative function

TL;DR

This paper establishes sharp almost sure bounds for weighted sums of Steinhaus random multiplicative functions, confirming predictions based on the law of the iterated logarithm.

Contribution

It provides the first rigorous proof of almost sure bounds for these sums, matching the predicted asymptotic behavior.

Findings

Sharp upper and lower bounds for the sums are proven.

Results align with predictions from the law of the iterated logarithm.

The bounds are almost sure, holding with probability one.

Abstract

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.