Dirichlet series associated to cubic fields with given quadratic resolvent

Abstract

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q. In a companion paper we do the same for quartic fields having a given cubic resolvent. As an application (not present in the initial version), we compute tables of the number of S_3-sextic fields E with |Disc(E)| < X, for X ranging up to 10^23. An accompanying PARI/GP implementation is available from the second author's website.