Dirichlet series associated to cubic fields with given quadratic resolvent

TL;DR

This paper derives explicit formulas for Dirichlet series counting cubic fields with a specified quadratic resolvent over Q, extending previous work and providing computational data for fields with discriminant bounds up to 10^23.

Contribution

It provides explicit formulas for Dirichlet series associated to cubic fields with a given quadratic resolvent, advancing the analysis from prior general character sum approaches.

Findings

Explicit formulas for Dirichlet series over Q for cubic fields with quadratic resolvent.

Computed tables of S_3-sextic fields with discriminant less than 10^23.

Implementation available in PARI/GP for further research.

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.