Partial sums of biased random multiplicative functions

Abstract

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 Ef(p)<0, ∀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). We establish a number of necessary and sufficient conditions for Mf(x)=o(x1−α) for some α>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) for all ϵ>0a.s., for each α>0, where {fα}α is a certain family of weakly biased random multiplicative functions.