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 ζ(s)=∑n≥11∕ns, s>1, then we consider the random variable Xs with P(Xs=n)=1∕(ζ(s)ns), n≥1. We derive various results such as the analog of the Erdös–Kac central limit theorem (CLT) for the number of distinct prime factors, ω(Xs), of Xs, as s↘1, large and moderate deviations for ω(Xs), and a Berry–Esseen result. In addition, we prove analogs of Erdös–Delange type formulas for expectations of additive and multiplicative functions evaluated at Xs. 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,…,N} or according to the harmonic distribution from their analogous asymptotic results for Xs as s↘1.