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The mean value of the product of class numbers of paired quadratic fields. I. (English) Zbl 1020.11079
This paper is the first of three papers, but one should consider Parts I and II [J. Math. Soc. Japan 55, No. 3, 739–764 (2003; Zbl 1039.11087)] together as the authors point out. The main results of this paper are (1) the conditions by which the filtering process works are identified and (2) the mean value of the product of class numbers of paired quadratic fields exists and is explicitly described as an Euler product. The local factors of this Euler product are related to the orbital Igusa local zeta-function of some prehomogeneous vector spaces and their explicit formulae are given in Part II and III [J. Number Theory 99, No. 1, 185–218 (2003; Zbl 1039.11086)].

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