Random multiplicative functions: the Selberg-Delange class

Abstract

Let $1∕2\le \mathit{\beta }<1$, p be a generic prime number and $f_{\mathit{\beta }}$ be a random multiplicative function supported on the squarefree integers such that ${(f_{\mathit{\beta }}(p))}_p$ is an i.i.d. sequence of random variables with distribution $\mathbb{P}(f(p)=-1)=\mathit{\beta }=1-\mathbb{P}(f(p)=+1)$. Let $F_{\mathit{\beta }}$ be the Dirichlet series of $f_{\mathit{\beta }}$. We prove a formula involving measure-preserving transformations that relates the Riemann ζ function with the Dirichlet series of $F_{\mathit{\beta }}$, 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 sum of $f_{\mathit{\beta }}$.