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A mean-value theorem for class numbers of quadratic extensions. (English) Zbl 0791.11058
Knopp, Marvin (ed.) et al., A tribute to Emil Grosswald: number theory and related analysis. Providence, RI: American Mathematical Society. Contemp. Math. 143, 179-242 (1993).
Let $$\omega(L)$$ denote the residue of the Dedekind zeta-function of a number field $$L$$. The author proves an asymptotic formula $\sum_{L\in A(S),\;D(L\mid K) \leq x} \omega(L) \sim C(K,S)x \qquad \text{ as } x \to \infty,$ where $$A(S)$$ denotes the set of the quadratic extensions $$L \mid K$$ of a number field $$K$$ with a fixed decomposition type for primes in a finite set $$S$$ (containing all the infinite primes of $$K$$), $$D(L\mid K)$$ is the norm of the relative discriminant of $$L\mid K$$; the constant $$C(K,S)$$ is given explicitly and may be compared with the main term of the asymptotic formula in [D. Goldfeld and J. Hoffstein, Invent. Math. 80, 185–208 (1985; Zbl 0564.10043)], where the case $$K = \mathbb Q$$ has been treated. Furthermore, he proves a Dirichlet series identity for a quadratic extension of a number field $$K$$, for which the class number $$h_ K$$ is odd; this identity may be viewed as a generalisation of the classical Gauss identity relating the class number of an order in a quadratic number field $$\mathbb Q(\sqrt{D})$$ to the class number of the field.
These results are obtained as a consequence of the adelic theory of zeta- functions associated to the space of binary quadratic forms. Giving a credit for developing this theory to A. Yukie [Math. Ann. 292, 355–394 (1992; Zbl 0757.11027)], the author chooses, however, to give an independent treatment of the theory.
For the entire collection see [Zbl 0773.00030].
Reviewer: B.Z.Moroz (Bonn)

##### MSC:
 11R29 Class numbers, class groups, discriminants 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11E41 Class numbers of quadratic and Hermitian forms 11M41 Other Dirichlet series and zeta functions 11R11 Quadratic extensions