Voronin Universality in several complex variables

TL;DR

This paper extends the Voronin universality theorem to multiple Hurwitz zeta-functions in several complex variables, demonstrating their universality and providing the first example of a multi-variable universal Dirichlet series.

Contribution

It proves the Voronin universality theorem for multiple Hurwitz zeta-functions in several variables, answering a longstanding question and establishing the first multi-variable universal Dirichlet series.

Findings

Multiple Hurwitz zeta-functions are universal in several complex variables.

The Euler-Zagier multiple zeta-function is shown to be universal in multiple variables.

First example of a Dirichlet series with universality in more than one variable.

Abstract

We prove the Voronin universality theorem for the multiple Hurwitz zeta-function with rational or transcendental parameters in $\mathbb{C}^n$ answering a question of Matsumoto. In particular this implies that the Euler-Zagier multiple zeta-function is universal in several complex variables and gives the first example of a Dirichlet series that is universal in more than one variable.