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