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We investigate the topological zeta function for unimodal maps in general anddynamical zeta functions for the tent map in particular. For the genericsituation, when the kneading sequence is aperiodic, it is shown that the zetafunctions have a natural boundary along its radius of convergence, beyond whichthe function lacks analytic continuation. We make a detailed study of thefunction $\prod_{n=0}^{\infty}(1-z^{2^n})$ associated with sequences of perioddoublings. It is demonstrated that this function has a dense set of poles andzeros on the unit circle, exhibiting a rich number theoretical structure.Comment: 12 pages LaTe

On the effect of pruning on the singularity structure of zeta functions