  In this paper we treat the classical Riemann zeta function as a function of
three variables: one is the usual complex $\adyn$-dimensional, customly denoted
as $s$, another two are complex infinite dimensional, we denote it as $\b =
\{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b =
\{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the
usual Riemann zeta function. Our goal in this paper is to study the meromorphic
continuation of $\zeta (\b , \z ,s)$ as a function of the triple $(\a , \z ,
s)$. Minor corrections, to appear in the Journal of Mathematical Analysis and
Applications.
