Multidimensional Shintani Zeta Functions and Zeta Distributions on $\mathbb{R}^d$

Abstract

The class of Riemann zeta distribution is one of the classical classes of probability distributions on $\mathbb{R}$. Multidimensional Shintani zeta function is introduced and its definable probability distributions on $\mathbb{R}^d$ are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on $\mathbb{R}^d$, is also studied.