Almost sure bounds for weighted sums of Rademacher random multiplicative functions

TL;DR

This paper establishes almost sure bounds for weighted sums of Rademacher random multiplicative functions, revealing their typical growth and large deviations, and compares these results to the Steinhaus case.

Contribution

It provides new almost sure bounds for Rademacher multiplicative functions and highlights differences from the Steinhaus case, especially regarding multiplicative chaos.

Findings

Sum of f(n)/√n is typically bounded by (log log x)^{3/4+ε}.

Existence of large deviations where sum exceeds (log log x)^{-1/2}.

Sharper bounds when restricting to integers with large prime factors.

Abstract

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily large values of $x$ such that $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \gg (\log\log(x))^{-1/2}$. This is different to what is found in the Steinhaus case, this time with the size of the Rademacher Euler product making the multiplicative chaos contribution the dominant one. We also find a sharper upper bound when we restrict to integers with a prime factor greater than $\sqrt{x}$, proving that $\sum_{\substack{n \leqslant x \\ P(n) > \sqrt{x}}}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{1/4+\epsilon}$.