A COMPOUND POISSON DISTRIBUTION WITH RIEMANN ZETA SUMMANDS: CHARACTERIZATION AND INFINITE DIVISIBILITY

Abstract

Infinitely divisible distributions constitute an important class of probability distributions due to their fundamental role in convolution structures, limit theorems, and Lévy processes. The Riemann zeta distribution, obtained from the Riemann zeta function through the transformation (X = log n), is a discrete infinitely divisible distribution whose properties are closely related to the von Mangoldt function. Motivated by this connection, this paper investigates a compound Poisson distribution generated by a random sum of independent and identically distributed Riemann zeta random variables. The characteristic function of the resulting distribution is derived using the Fourier–Stieltjes transform. By constructing an auxiliary family of functions satisfying the necessary properties of characteristic functions, we prove the infinite divisibility of the proposed compound distribution. Furthermore, we show that the corresponding characteristic function is uniformly continuous, positive definite, and has no real zeros. These results provide a number-theoretic extension of existing characterizations of compound Poisson distributions
with other infinitely divisible summands, including geometric and Cauchy-variance distributions.