Many sources refer to zeta functions as a kind of generating function, but I have to admit: I have never fully understood what this means. (Example of such language: the first two sentences in this book read, ``Zeta functions encode the counting of certain objects of geometric, algebraic, or arithmetic behavior. What distinguishes them from other generating series are special analytic or algebraic properties.'')
In some cases, there is a clear connection: The Ihara zeta function of a graph is the exponential of the generating function for the sequence $N_l / l$, where $l$ is a natural number and $N_l$ is the number of closed cycles of length $l$ in the graph; the zeta function of a variety defined over a finite field is defined the same way, except with $N_l$ being the number of points in $\mathbb{F}_{p^l}$.
But what about the zeta function? Is there a way of writing $\zeta(s)$ as the exponential of some generating series?