Partial sums of biased random multiplicative 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.