Let 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∈P form a sequence of ±1 valued independent random variables with e6d68f810d84d4ecea3d51aaa8026" title="Click to view the MathML source">Ef(p)<0, ∀p∈P. The function f is called strongly biased (towards classical Möbius function), if e529e624c34f6432">a.s. , and it is weakly biased if converges a.s. Let e62eb7abe73e525d81ca996b97e725c" title="Click to view the MathML source">Mf(x):=∑n≤xf(n). We establish a number of necessary and sufficient conditions for Mf(x)=o(x1−α) for some 82d" title="Click to view the MathML source">α>0, a.s., when f is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if e50cc2696ff8afbb" title="Click to view the MathML source">Mfα(x)=o(x1/2+ϵ) for all ϵ>0a.s. , for each 82d" title="Click to view the MathML source">α>0, where a4bf09c8e701c25" title="Click to view the MathML source">{fα}α is a certain family of weakly biased random multiplicative functions.