  The primary purpose of this paper is to generalize the classical Riemann zeta
function to the setting of Krull monoids with torsion class groups. We provide
a first study of the same generalization by extending Euler's classical product
formula to the more general scenario of Krull monoids with torsion class
groups. In doing so, the Decay Theorem is fundamental as it allows us to use
strong atoms instead of primes to obtain a weaker version of the Fundamental
Theorem of Arithmetic in the more general setting of Krull monoids with torsion
class groups. Several related examples are exhibited throughout the paper, in
particular, algebraic number fields for which the generalized Riemann zeta
function specializes to the Dedekind zeta function.
