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The mean value of the product of class numbers of paired quadratic fields. II. (English) Zbl 1039.11087
For Part I, cf. Tôhoku Math. J. (2) 54, 513–565 (2002; Zbl 1020.11079).
In this second part of a series of papers the authors continue their study on the explicit computations of the mean value of the product of class numbers and regulators of two quadratic extensions $$F$$, $$F^*\neq\widetilde k$$ contained in the biquadratic extensions of $$k\subset\widetilde k$$. Let $$k$$ be a number field, let $$\Delta_k$$, $$h_k$$ and $$R_k$$ be the absolute discriminant, which is an integer, the class number and the regulator, respectively. We fix a number field $$k$$ and a quadratic extension $$\widetilde k$$ of $$k$$. If $$F\neq\widetilde k$$ is another quadratic extension of $$k$$, let $$\widetilde F$$ be the composite of $$F$$ and $$\widetilde k$$. Then $$\widetilde F$$ is a biquadratic extension of $$k$$ and so contains precisely three quadratic extensions, $$\widetilde k$$, $$F$$ and the third one $$F^*$$ of $$k$$. $$F$$ and $$F^*$$ are said to be paired. The main theorem of this series of papers are the following two results: (1) With either choice of sign we have $\lim_{X\to\infty} X^{-2} \sum_{[F:\mathbb Q]= 2,\;0<\pm\Delta_F< X} h_F R_Fh_{F^*} R_{F^*}= c_{\pm}(d_0)^{-1} M(d_0).$ (2) With either choice of sign we have $\lim_{X\to \infty} X^{-2} \sum_{[F:\mathbb Q]= 2,\;0< \pm\Delta_F< X} h_{F( \sqrt{d_0})} R_{F(\sqrt{d_0})}= c_{\pm}(d_0)^{-1} h_{\mathbb Q(d_0)} R_{\mathbb Q(d_0)} M(d_0).$ Here $$M(d_0)$$ is a number-theoretical quantity like an Euler product. In the second part, the authors compute the local density for all non-dyadic cases and for those dyadic cases that can be dealt with in a similar manner.

##### MSC:
 11R45 Density theorems 11R29 Class numbers, class groups, discriminants 11S90 Prehomogeneous vector spaces 11S40 Zeta functions and $$L$$-functions