After a discussion of Hasse-Weil zeta functions of arithmetic varieties X, beginning with the Riemann zeta function, we explain Lichtenbaum's conjecture for their values at zero. We then discuss a class of dynamical systems whose closed orbits behave in some sense like the closed points of an arithmetic variety. Under some extra conditions we show that using the Cheeger-Müller secondary index theorem an analogue of Lichtenbaum's conjecture can be proved for their Ruelle zeta functions. Moreover, for any arithmetic variety X, we sketch the construction of a dynamical system whose periodic orbits correspond to the closed points of X. In particular, for X = spec Z we obtain a dynamical system whose periodic orbits correspond to the prime numbers p, with the length of the orbit being log p.