Duke Mathematical Journal

Error estimates for the Davenport-Heilbronn theorems

Karim Belabas, Manjul Bhargava, and Carl Pomerance

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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.

Article information

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

First available in Project Euclid: 28 April 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R11: Quadratic extensions
Secondary: 11R29: Class numbers, class groups, discriminants 11R45: Density theorems


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

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