  Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random
multiplicative function supported on the squarefree integers such that
$(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution
$\mathbb{P}(f(p)=-1)=\beta=1-\mathbb{P}(f(p)=+1)$. Let $F_\beta$ be the
Dirichlet series of $f_\beta$. We prove a formula involving measure-preserving
transformations that relates the Riemann $\zeta$ function with the Dirichlet
series of $F_\beta$, for certain values of $\beta$, and give an application.
Further, we prove that the Riemann hypothesis is connected with the mean
behavior of a certain weighted partial sums of $f_\beta$.
