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Moments of class numbers of binary quadratic forms with algebraic integer coefficients. (Momente der Klassenzahlen binĂ¤rer quadratischer Formen mit ganzalgebraischen Koeffizienten.) (German) Zbl 0817.11025
Let $$k$$ be a totally real number field with odd ideal class number, $$\sigma_ 1, \dots, \sigma_ n: k\to \mathbb{R}$$ the embeddings of $$k$$ in $$\mathbb{C}$$, and $${\mathcal O}_ k$$ the ring of integers in $$k$$ with the group of units $$E({\mathcal O}_ k)$$. Define in $$\widetilde {\mathcal D}:= \{D\in {\mathcal O}_ k\mid x^ 2\equiv D(4)$$ has a solution in $${\mathcal O}_ k$$, $$D$$ no square in $${\mathcal O}_ k$$, $$\sigma_ 1(D) >0, \sigma_ 2<0, \dots, \sigma_ n(D)<0\}$$, the equivalence relation $D_ 1\sim D_ 2: \iff D_ 1= e^ 2 D_ 2 \quad \text{with} \quad e\in E({\mathcal O}_ k).$ Let $${\mathcal D} \subset \widetilde {\mathcal D}$$ be a full system of representatives of $$\sim$$. For $$D\in \widetilde {\mathcal D}$$ it is proved that the group $U_ D:= \bigl\{ {\textstyle {1\over 2}} (x+ y\sqrt {D}) \mid x,y\in {\mathcal O}_ k,\;x^ 2- Dy^ 2=4 \bigr\}$ has a direct decomposition $$U_ D= \{\pm1\} \times \langle \varepsilon_ D \rangle$$ with unique fundamental unit $$\varepsilon_ D>1$$.
If $$D\in {\mathcal O}_ k$$, $$x^ 2\equiv D(4)$$ has a solution in $${\mathcal O}_ k$$, $$a,b,c\in {\mathcal O}_ k$$, $$a{\mathcal O}_ k+ b{\mathcal O}_ k+ c{\mathcal O}_ k= {\mathcal O}_ k$$, $$D= b^ 2- 4ac$$, then the polynomial $$ax^ 2+ bxy+ cy^ 2$$ is called a primitive quadratic form with discriminant $$D$$. Equivalence of quadratic forms is defined in the usual way via linear transformations of variables with matrix $$\left( \begin{smallmatrix} p &q\\ r &s\end{smallmatrix} \right)$$, $$p,q,r,s\in {\mathcal O}_ k$$, $$ps- qr=1$$. The number of equivalence classes is denoted by $$h(D)$$. A Dirichlet formula connecting $$h(D)$$ with the value of a certain $$L$$- series at $$s=1$$ is proved along the usual lines. With this the following formula is proved for all $$m\in \mathbb{N}$$ and $$\varepsilon>0$$ $\sum_{D\in {\mathcal D}: \varepsilon_ D\leq x} (h(D)\log \varepsilon_ D )^ m= \lambda_ m x^{m+1}+ O(x^{m+ \rho+ \varepsilon})$ with explicit $$0<\rho <1$$. This generalizes work of P. Sarnak [J. Number Theory 15, 229-247 (1982; Zbl 0499.10021) and ibid. 21, 333-346 (1985; Zbl 0571.10022); Acta Math. 151, 253-295 (1983; Zbl 0527.10022)]. The use of the Selberg zeta function is avoided. Instead classical density estimates for the zeros of Dirichlet $$L$$-functions and the method of M. B. Barban [Russ. Math. Surv. 21, 49-103 (1966); translation from Usp. Mat. Nauk 21, No. 1, 51-102 (1966; Zbl 0234.10031)] are used.

##### MSC:
 11E41 Class numbers of quadratic and Hermitian forms 11R29 Class numbers, class groups, discriminants 11E16 General binary quadratic forms 11R42 Zeta functions and $$L$$-functions of number fields
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