Partial sums of biased random multiplicative functions

Let PP 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∈P{f(p)}p∈P form a sequence of ±1 valued independent random variables with Ef(p)<0Ef(p)<0, ∀p∈P∀p∈P. The function f   is called strongly biased (towards classical Möbius function), if ∑p∈Pf(p)p=−∞a.s.  , and it is weakly biased if ∑p∈Pf(p)p converges a.s.   Let Mf(x):=∑n≤xf(n)Mf(x):=∑n≤xf(n). We establish a number of necessary and sufficient conditions for Mf(x)=o(x1−α)Mf(x)=o(x1−α) for some α>0α>0, a.s., when f   is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if Mfα(x)=o(x1/2+ϵ)Mfα(x)=o(x1/2+ϵ) for all ϵ>0ϵ>0a.s.  , for each α>0α>0, where {fα}α{fα}α is a certain family of weakly biased random multiplicative functions.