The mean value of the product of class numbers of paired quadratic fields. II. (Q1411761)

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.