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On proportional constants of the mean value of class numbers of quadratic extensions. (English) Zbl 1134.11041
Let \(k\) be an algebraic number field and \(S\) be a finite set of places of \(k\) containing all infinite places. The author has given a refinement of a result of R. Datskovsky [Contemp. Math. 143, 179–242 (1993; Zbl 0791.11058)] concerning the asymptotic behavior of \[ \lim_{X\to\infty}\, X^{-{3\over 2}}\sum_F h_F R_F, \] where \(F\) runs over quadratic extensions of \(k\) satisfying \(N(\Delta_{F/k})\leq X\) and the prefixed splittting data at place in \(S\), \(N(\Delta_{F/k})\) is the absolute norm of the discriminant of \(F_k\), and \(h_F\) and \(R_F\) are the class number and the regulator of \(F\). The limit can be expressed by arithmetic data of \(k\) and \(S\); \[ c_k\prod_{v\in S} e_v\prod_{v\not\in S} f_v, \] where \(c_k\) is a constant determined by \(k\) involving \(h_k\), \(R_k\) and \(\zeta_k(2)\) as factors, \(e_v\) is a conststant depending on the fixed splitting data \(F_v\) for \(v\in S\), and \(f_v= 1- q^{-2}_v- q^{-3}_v+ q^{-4}_v\) for \(v\not\in S\).
The author has given the explicit value of \(e_v\) for which \(F_v/k_v\) is a ramified quadratic extension, which was not determined in the paper of Datskovsky.

11R45 Density theorems
11S90 Prehomogeneous vector spaces
Full Text: DOI arXiv
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