For that purpose, it is most efficient to use the well-known relation between residues of zeta functions and asymptotic expansions of related theta functions.
Then, their asymptotic behavior is shown to be related to the singularities on the critical line of twisted zeta functions.
This approach also allows computation of the zeta function for the substitution.
We believe it sheds a new light on weighted dynamical zeta functions associated to smooth (non-analytic) maps.
The zeta function will clearly be the same for functions which differ by coboundaries or which are related by automorphisms.
Moreover, there are no assumptions on the zeta functions of the systems under consideration.
The example above clearly exhibits this phenomenon for the parameterization of zeta functions.
Therefore, the following parameterization of zeta functions is obtained.
