Suppose I have a discrete dynamical system $f : X \to X$, for some metric space $X$. The zeta function of this system is defined as $$\zeta_f(z) = \exp \sum_{n=1}^\infty \frac 1 n \left| \mathrm{Fix}(f^n)\right| z^n,$$ and the generating function is $$g_f(z) = \sum_{n=1}^\infty \left| \mathrm{Fix}(f^n)\right| z^n,$$ where $\mathrm{Fix}(f^n) = \left\{ x \in X : f^n(x) = x\right\}$ and $|\mathrm{Fix}(f^n)|$ is the cardinality (assume this is finite for every $n$). I am trying to prove Exercise 3.4.7 from Brin and Stuck's "Introduction to Dynamical Systems":
Prove that if the zeta function is rational, then so is the generating function.
I can show that there are different examples of systems with rational zeta function (namely subshifts of finite type and hyperbolic toral automorphisms), and the generating functions of these examples also happen to be rational, but I'm not sure how to prove this in general. I think I'm missing some power series identity that will make this easier but I'm not sure which. Any suggestions?
Since $(\log \zeta_f(z) )' =\dfrac{\zeta_f'(z) }{\zeta_f(z) } =\dfrac{g_f(z)}{z} $, if $\zeta_f(z)$ is rational then $\zeta_f'(z)$ is rational so $g_f(z) =\dfrac{z\zeta_f'(z) }{\zeta_f(z) } $ is rational.