Error estimates for the Davenport-Heilbronn theorems
by
Carl Pomerance
Mathematics Department, Dartmouth College
Coauthors: Karim Belabas and Manjul Bhargava

We improve the known remainder estimate in the theorem of Davenport and Heilbronn which counts the number of cubic fields with discriminant up to a given bound. The method works also to improve the error estimate for their theorem on the average 3-torsion in the class group of quadratic fields. In particular, our improved error estimates allow the analytic continuation of the related Dirichlet series to the left of the 1-line, thus answering a question of Henri Cohen.

Date received: April 16, 2004