Random multiplicative functions: The Selberg-Delange class

TL;DR

This paper explores the properties of random multiplicative functions supported on squarefree integers, establishing a connection between their Dirichlet series and the Riemann zeta function, and linking the Riemann hypothesis to their mean behavior.

Contribution

It introduces a measure-preserving transformation relating the Riemann zeta function to the Dirichlet series of random multiplicative functions, and connects the Riemann hypothesis with their partial sums.

Findings

Derived a formula linking the zeta function with the Dirichlet series of $f_eta$

Established a connection between the Riemann hypothesis and the mean behavior of weighted partial sums of $f_eta$

Provided an application of the measure-preserving transformation in number theory

Abstract

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 $f_\beta$.