The Ruelle Zeta Function (RZF) is defined in analogy to the Riemann Zeta Function, where primes correspond to primitive closed orbits of an Anosov flow X. The RZF extends meromorphically to the whole complex plane and carries rich information about the flow. Using microlocal methods, Dyatlov-Zworski recently showed that the order of vanishing at zero n(X) of the RZF equals the minus Euler characteristic, if X is the geodesic vector field of a negatively curved surface. In this talk, I will explain an exciting novel result showing the instability of n(X) close to hyperbolic 3-manifolds, starkly contrasting the case of surfaces. The proof is based on studying the pushforward of a certain pairing between resonant states (“eigenstates of X”), regularisation arguments and wavefront set calculus. Joint work with Semyon Dyatlov, Benjamin Küster and Gabriel Paternain.