On properties of the Riemann zeta distribution

Abstract

We examine various properties of positive integers selected according to the Riemann zeta distribution. That is, if $\zeta (s)={\sum }_{n\ge 1}1∕n^s$, $s>1$, then we consider the random variable $X_s$ with $P(X_s=n)=1∕(\zeta (s)n^s)$, $n\ge 1$. We derive various results such as the analog of the Erdös–Kac central limit theorem (CLT) for the number of distinct prime factors, $\omega (X_s)$, of $X_s$, as $s\searrow 1$, large and moderate deviations for $\omega (X_s)$, and a Berry–Esseen result. In addition, we prove analogs of Erdös–Delange type formulas for expectations of additive and multiplicative functions evaluated at $X_s$. We also examine various applications using Dirichlet series. Finally, we show how to derive asymptotic distributional results for an integer selected uniformly at random from $[N]=\{1,2,\dots ,N\}$ or according to the harmonic distribution from their analogous asymptotic results for $X_s$ as $s\searrow 1$.