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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.

##### MSC:
 11R45 Density theorems 11S90 Prehomogeneous vector spaces
##### Keywords:
density theorems; prehomogeneous vector spaces
Full Text:
##### References:
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