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Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

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Zeta function for curves in a manifold

There's been a lot of work since Smale's idea of a dynamical zeta function for general flows (in particular geodesic flows). A good starting point would be this 12 year old review by Baladi. There is …
Thomas Sauvaget's user avatar