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Moments of class numbers of binary quadratic forms with algebraic integer coefficients. (Momente der Klassenzahlen binärer 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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