Error estimates for the Davenport-Heilbronn theorems

Abstract

We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of $3$-torsion elements in the class groups of quadratic fields. In addition, we prove analogous error terms for the density of discriminants of quartic fields and the mean number of $2$-torsion elements in the class groups of cubic fields. These results prove analytic continuation of the related Dirichlet series to the left of the line $R(s)=1$.