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 -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 -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 .
Karim Belabas. Manjul Bhargava. Carl Pomerance. "Error estimates for the Davenport-Heilbronn theorems." Duke Math. J. 153 (1) 173 - 210, 15 May 2010. https://doi.org/10.1215/00127094-2010-007