Error estimates for the Davenport-Heilbronn theorems
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 \[ \sum_{0<-\text{Disc}(K_3)<X}\# \text{Cl}_2(K_3)=\frac 32\sum_{0<-\text{Disc}(K_3)<X}1+O(X^{\frac {23}{24}+\varepsilon}). \]
scientific article; zbMATH DE number 5717814
scientific article