Partial sums of biased random multiplicative functions

TL;DR

This paper investigates the behavior of partial sums of biased random multiplicative functions supported on squarefree integers, establishing conditions for their growth and linking the Riemann Hypothesis to their asymptotic properties.

Contribution

It provides necessary and sufficient conditions for the growth of partial sums of biased random multiplicative functions and connects these results to the Riemann Hypothesis.

Findings

Conditions for $M_f(x)=o(x^{1-eta})$ for biased functions

Characterization of strong and weak bias in random multiplicative functions

Equivalence of the Riemann Hypothesis to bounds on partial sums of certain weakly biased functions

Abstract

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random variables with $\mathbb{E} f(p)<0$, $\forall p\in \mathcal{P}$. The function $f$ is called strongly biased (towards classical M\"obius function), if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p}=-\infty$ a.s., and it is weakly biased if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p} $ converges a.s. Let $M_f(x):=\sum_{n\leq x}f(n)$. We establish a number of necessary and sufficient conditions for $M_f(x)=o(x^{1-\alpha})$ for some $\alpha>0$, a.s., when $f$ is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if $M_{f_\alpha}(x)=o(x^{1/2+\epsilon})$ for all $\epsilon>0$ a.s., for each $\alpha>0$, where $\{f_\alpha \}_\alpha$ is a certain family of weakly biased random multiplicative functions.