A Billiard in an Open Circle and the Riemann Zeta Function

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We consider a dynamical billiard in a circle with one or two holes in the boundary, or q symmetrically placed holes. It is shown that the long-time survival probability, either for a circle billiard with discrete or with continuous time, can be written as a sum over never-escaping periodic orbits. Moreover, it is demonstrated that in both cases the Mellin transform of the survival probability with respect to the hole size has poles at locations determined by zeros of the Riemann zeta function and, in some cases, Dirichlet L functions.