Dirichlet Series Associated to Cubic Fields with Given Quadratic Resolvent

Let k be a quadratic field. We give an explicit formula for the Dirichlet series Sigma(K) vertical bar Disc(K)vertical bar(-s), where the sum is over isomorphism classes of all cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to [14] (see also [22]), 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, and in a companion paper we do the same for quartic fields having a given cubic resolvent. As an application, we compute tables of the number of S-3-sextic fields E with vertical bar Disc(E)vertical bar < X for X ranging up to 10(23). An accompanying PARI/GP implementation is available from the second author's website.