Behaviors of multivariable finite Euler products in probabilistic view

TL;DR

This paper investigates the probabilistic properties of multivariable finite Euler products, revealing their diverse behaviors in generating various types of probability distributions, including infinitely divisible and non-characteristic functions.

Contribution

It introduces an analysis of multivariable finite Euler products' behavior concerning their probability distribution properties, expanding understanding beyond classical zeta functions.

Findings

Finite Euler products can generate a range of probability distributions.

Some multivariable Euler products produce infinitely divisible characteristic functions.

Others generate functions that are not characteristic functions.

Abstract

Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include functions which generate infinitely divisible, not infinitely divisible characteristic functions and not even to be characteristic functions. In this paper, we treat some multivariable finite Euler products and show how they behave in view of such properties.