A Billiard in an Open Circle and the Riemann Zeta Function 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.

"A Billiard in an Open Circle and the Riemann Zeta Function" by Leonid A. Bunimovich and Carl P. Dettmann. #ExperimentalMath #RiemannZetaFunction #MathSky

Level Curves for Zhang’s Eta Function Study of the level curve Re(\eta(s))=0 for \eta(s)=\pi^{s/2}\Gamma(s/2)\zeta'(s) gives a new classification of the zeros of \zeta(s) and of \zeta'(s). We conjecture that for type 2 zeros, liminf(\beta'-1/2)log \gamma'=0 if and only if liminf(\gamma^+ - \gamma^-)log \gamma'=0, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify 10^6 zeros of \zeta'(s) near T=10^10. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

"Level Curves for Zhang’s Eta Function" by Jeffrey Stopple. #ExperimentalMath #NumberTheory #RiemannZetaFunction #MathSky

#mathematics #math #NumberTheory #RiemannZetaFunction
Source:https://buff.ly/40vMNTS