For a large class of open hyperbolic dynamical systems, we introduce a weighted zeta function. Using microlocal analysis we can prove its meromorphic continuation and prove that the residues are given by the so called “invariant Ruelle distributions”. In the particular case of Schottky surfaces we will see, that these residue formulas for the basis for an efficient numerical algorithm to calculate the invariant Ruelle distributions and conjecturally also quantum phase space distributions.