  We investigate the topological zeta function for unimodal maps in general and
dynamical zeta functions for the tent map in particular. For the generic
situation, when the kneading sequence is aperiodic, it is shown that the zeta
functions have a natural boundary along its radius of convergence, beyond which
the function lacks analytic continuation. We make a detailed study of the
function $\prod_{n=0}^{\infty}(1-z^{2^n})$ associated with sequences of period
doublings. It is demonstrated that this function has a dense set of poles and
zeros on the unit circle, exhibiting a rich number theoretical structure.
