Random multiplicative functions: The Selberg-Delange class

Abstract

Let $1/2\le \beta <1$, $p$ 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 $f_{\beta }$.

9 pages