The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers $p$.
I am interest in analogous versions in the theory of dynamical systems:
Discrete time. For a map $f\colon X\to X$ we define a zeta function by $$ \zeta_f(s)=\exp\sum_{k=1}^\infty\frac{n_k}k s^k, $$ where $n_k$ is the number of fixed points of $f^k$. For example, if $f$ is the Markov chain $\sigma_A$ with transition matrix $A$, we have $n_k={\rm tr} (A^k)$ and so, after some computations, $$ \zeta_f(s)=\exp\sum_{k=1}^\infty\frac{{\rm tr} (A^k)}k s^k=\frac1{\det(I-s A)}. $$
Continuous time. For the geodesic flow on (the unit tangent bundle of) a compact surface $V$ with constant curvature $-1$ we define a zeta function by $$ \zeta_V(s)=\prod_\gamma\frac1{1-e^{-s l(\gamma)}}, $$ where the product is over all closed geodesics $\gamma$ and where $l(\gamma)$ is the length of $\gamma$ (there are countably many closed geodesics, one in each conjugacy class of the fundamental group of $V$). Notice that $\zeta_V(s)=Z(s+1)/Z(s)$, where $Z$ is the Selberg zeta function.
Question 1: Is there an alternative formula for $\zeta_V(s)$ that imitates the identity $$ \sum_{n=1}^\infty \frac1{n^s}=\prod_p\frac1{1-p^{-s}}? $$
Of course, one can also consider the case of variable negative curvature, but this seemed to be a good starting point.
Question 2: Is there an alternative formula for $\zeta_f(s)$ and an arbitrary map $f$ that could lead to a similar identity?