The Riemann Zeta Distribution

Abstract

Let ζ be the Riemann zeta function. Khinchine (1938) proved that the function f<sub>σ</sub>(t)=ζ(σ+i t)/ζ(σ), where σ> 1 and t is real, is an infinitely divisible characteristic function. We investigate further the fundamental properties of the corresponding distribution of f<sub>σ</sub>, the Riemann zeta distribution, including its support and unimodality. In particular, the Riemann zeta random variable is represented as a linear function of infinitely many independent geometric random variables. To extend Khinchine's result, we construct the Dirichlet-type characteristic functions of discrete distributions and provide a sufficient condition for the infinite divisibility of these characteristic functions. By way of applications, we give probabilistic proofs for some identities in number theory, including a new identity for the reciprocal of the Riemann zeta function.