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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)

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