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On evaluation of zeta functions of totally real algebraic number fields at non-positive integers. (English) Zbl 0349.12007
Let \(F\) be a totally real algebraic number field of degree \(n\). If \(\mathbf b\) and \(\mathbf f\) are mutually prime integrals of \(F\), let \(\zeta(\mathbf b,\mathbf f,s)=\sum N(\mathbf a)^{-s}\), where the summation is over all integral ideals \(\mathbf a\) of \(F\) which are in the same narrow ray class mod \(\mathbf f\) as \(\mathbf b\). H. Klingen [Math. Ann. 145, 265–272 (1962; Zbl 0101.03002)] and C. L. Siegel [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1969, 87–102 (1969; Zbl 0186.08804)] have determined the values of \(\zeta(\mathbf b,\mathbf f,s)\) at non-positive integers. Their methods depend on the theory of elliptic modular forms.
In this interesting paper the author introduces a further relatively simple and straightforward method for determining these values. It is based on an \(n\)-dimensional generalization of the classical Hankel type integral for the zeta-function. An essential point in the argument is the partition of \(\mathbb R^n\) into a disjoint union of “open simplicial cones”, i.e. subsets of the form \(\{x_1v_1+\dots+x_rv_r\mid x_i>0\}\) for any set of linearly independent vectors \(v_1,\dots,v_r\). An application of the result to continued fractions of quadratic irrationalities is given. This extends a previous result of F. Hirzebruch [Enseign. Math., II. Sér. 19, 183–281 (1973; Zbl 0285.14007)]. Let \(K\) be a totally imaginary quadratic extension of \(F\) and let \(H\) and \(h\) denote the class numbers of \(K\) and \(F\), respectively. The author derives a formula for \(H/h\) which may be regarded as an affirmative answer to the Hecke conjecture that \(H/h\) admits an elementary arithmetic expression. Another such formula has been previously given by L. J. Goldstein [Manuscr. Math. 9, 245–305 (1973; Zbl 0259.12006)]; cf. also L. J. Goldstein and P. de la Torre [Nagoya Math. J. 59, 169–198 (1975; Zbl 0335.10031)].

11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11R42 Zeta functions and \(L\)-functions of number fields
11R80 Totally real fields
11R27 Units and factorization
11F55 Other groups and their modular and automorphic forms (several variables)