Zeta distributions generated by multidimensional polynomial Euler products with complex coefficients

TL;DR

This paper characterizes when multidimensional polynomial Euler products with complex coefficients generate various types of distributions, providing examples and exploring applications in analytic number theory.

Contribution

It offers necessary and sufficient conditions for these Euler products to produce infinitely divisible, quasi-infinitely divisible, or non-characteristic functions, advancing understanding of zeta distributions.

Findings

Conditions for generating infinitely divisible distributions

Examples of zeta distributions from polynomial Euler products

Applications to analytic number theory

Abstract

In the present paper, we treat multidimensional polynomial Euler products with complex coefficients on ${\mathbb{R}}^d$. We give necessary and sufficient conditions for the multidimensional polynomial Euler products to generate infinitely divisible, quasi-infinitely divisible but non-infinitely divisible or not even characteristic functions by using Baker's theorem. Moreover, we give many examples of zeta distributions on ${\mathbb{R}}^d$ generated by the multidimensional polynomial Euler products with complex coefficients. Finally, we consider applications to analytic number theory.