Dirichlet series associated to quartic fields with given resolvent

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series ${\sum }_K\mathrm{\mid }\text{Disc}(K){\mathrm{\mid }}^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.