Multidimensional Shintani zeta functions and zeta distributions on Rᵈ

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The class of Riemann zeta distribution is one of the classical classes of probability distributions on R. Multidimensional Shintani zeta function is introduced and its definable probability distributions on Rᵈ 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 Rᵈ, is also studied.