## Duke Mathematical Journal

### 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 $\mathcal{R}(s)= 1$.

#### Article information

Source
Duke Math. J. Volume 153, Number 1 (2010), 173-210.

Dates
First available in Project Euclid: 28 April 2010

http://projecteuclid.org/euclid.dmj/1272480934

Digital Object Identifier
doi:10.1215/00127094-2010-007

Mathematical Reviews number (MathSciNet)
MR2641942

Zentralblatt MATH identifier
1227.11114

Subjects
Secondary: 11R29: Class numbers, class groups, discriminants 11R45: Density theorems

#### Citation

Belabas, Karim; Bhargava, Manjul; Pomerance, Carl. Error estimates for the Davenport-Heilbronn theorems. Duke Math. J. 153 (2010), no. 1, 173--210. doi:10.1215/00127094-2010-007. http://projecteuclid.org/euclid.dmj/1272480934.

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