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arXiv · 2009.09240

Random multiplicative functions: The Selberg-Delange class

Marco Aymone
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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 $f_\beta$.

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Marco Aymone. 2020-09-19. Random multiplicative functions: The Selberg-Delange class. https://arxiv.org/abs/2009.09240

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