Error estimates for the Davenport-Heilbronn theorems
type: Item
instance of: scholarly article
title: Error estimates for the Davenport-Heilbronn theorems
zbMATH Open document ID: 1227.11114
DOI: 10.1215/00127094-2010-007
author: Karim Belabas
author: Manjul Bhargava
author: Carl Pomerance
published in: Duke Mathematical Journal
publication date: 2010-06-07T00:00:00Z
review text: The authors prove: for any \(\varepsilon>0\), the number of isomorphism classes of cubic fields whose discriminant \(D\) satisfies \(0<D<X\) is \[ \frac{1}{12\zeta(3)}X+O(X^{\frac 78+\varepsilon}) \] and the number of isomorphism classes of cubic fields whose discriminant \(D\) satisfies \(0<-D<X\) is \[ \frac{1}{4\zeta(3)}X+O(X^{\frac 78+\varepsilon}). \] Let \(D\) denote the discriminant of a quadratic field, and let \(\text{Cl}_3(D)\) denote the 3-torsion subgroup of the ideal class group of \(D\). For any \(\varepsilon>0\), \[ \sum_{0<D<X}\# \text{Cl}_3(D)=\frac 43\sum_{0<D<X}1+O(X^{\frac 78+\varepsilon}) \] and \[ \sum_{0<-D<X}\# \text{Cl}_3(D)=2\sum_{0<-D<X}1+O(X^{\frac 78+\varepsilon}). \] For any \(\varepsilon>0\) and for \(i=0,1,2\), the number of isomorphism classes of \(S_4\)-quartic fields having \(4-2i\) real embeddings, \(2i\) complex embeddings, and absolute discriminant less than \(X\) is \[ \frac{\beta}{48}X+O(X^{\frac {23}{24}+\varepsilon})\,\text{if}\,i=0, \] \[ \frac{\beta}{8}X+O(X^{\frac {23}{24}+\varepsilon})\,\text{if}\,i=1, \] \[ \frac{\beta}{8}X+O(X^{\frac {23}{24}+\varepsilon})\,\text{if}\,i=2, \] where \(\beta=\prod_p(1+p^{-2}-p^{-3}-p^{-4})\). Let \(K_3\) denote a cubic field, and let \(\text{Cl}_2(K_3)\) denote the 2-torsion subgroup of its ideal class group. For any \(\varepsilon>0\), \[ \sum_{0<\text{Disc}(K_3)<X}\# \text{Cl}_2(K_3)=\frac 54\sum_{0<\text{Disc}(K_3)<X}1+O(X^{\frac {23}{24}+\varepsilon}) \] and \[ \sum_{0<-\text{Disc}(K_3)<X}\# \text{Cl}_2(K_3)=\frac 32\sum_{0<-\text{Disc}(K_3)<X}1+O(X^{\frac {23}{24}+\varepsilon}). \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 5717814
zbMATH Keywords: discriminant
zbMATH Keywords: density
zbMATH Keywords: quadratic field
zbMATH Keywords: cubic field
zbMATH Keywords: quartic field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2010-007
OpenAlex ID: W2049367305
Wikidata QID: Q101094331
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description: scientific article; zbMATH DE number 5717814