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. We also prove joint and discrete universality results in several complex variables.
Multiple zeta functions of Kaneko-Tsumura type and their values at positive integers
On a two-variable zeta function for number fields
Multiple Dirichlet series and moments of zeta and L-functions
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