In this section we will study our zeta functions for -graph systems coming from labeled graphs.
An important technical tool in studying the zeta function is a family of linear operators.
In the second step, we deduce its asymptotic behavior from the knowledge of the singularities of these zeta functions.
We work with the reciprocals of these zeta functions, and consider their power series expansions.
At this point we can pass from spectral properties of operators to analyticity properties of determinants and zeta functions.
In the second section we introduce a dynamical zeta function and derive some important properties of its analytic extension.
We survey some recent progress in the theory of dynamical zeta functions and explain its implications for counting problems.
Therefore, the following parameterization of zeta functions is obtained.
