An almost sure upper bound for random multiplicative functions on integers with a large prime factor

TL;DR

This paper establishes an almost sure upper bound on the sum of random multiplicative functions over integers with large prime factors, revealing the typical size of their fluctuations as numbers grow large.

Contribution

It provides a new almost sure upper bound for sums of random multiplicative functions over integers with large prime factors, advancing understanding of their fluctuation behavior.

Findings

Sum of f(n) over n with P(n) > sqrt(x) is bounded by sqrt{x} (log log x)^{1/4+ε} almost surely.

The result characterizes the typical size of fluctuations of random multiplicative functions.

The bound holds for both Rademacher and Steinhaus random multiplicative functions.

Abstract

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.