  In this article we prove meromorphic continuation of weighted zeta functions
in the framework of open hyperbolic systems by using the meromorphically
continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a
residue formula proving equality between residues of weighted zetas and
invariant Ruelle distributions. We combine this equality with results of
Guillarmou, Hilgert and Weich (2021) in order to relate the residues to
Patterson-Sullivan distributions. Finally we provide proof-of-principle results
concerning the numerical calculation of invariant Ruelle distributions for
3-disc scattering systems.
