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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 $\bold b$ and $\bold f$ are mutually prime integrals of $F$, let $\zeta(\bold b,\bold f,s)=\sum N(\bold a)^{-s}$, where the summation is over all integral ideals $\bold a$ of $F$ which are in the same narrow ray class mod $\bold f$ as $\bold b$. {\it H. Klingen} [Math. Ann. 145, 265--272 (1962; Zbl 0101.03002)] and {\it C. L. Siegel} [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1969, 87--102 (1969; Zbl 0186.08804)] have determined the values of $\zeta(\bold b,\bold f,s)$ 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 $\Bbb R^n$ 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 {\it 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 {\it L. J. Goldstein} [Manuscr. Math. 9, 245--305 (1973; Zbl 0259.12006)]; cf. also {\it L. J. Goldstein} and {\it P. de la Torre} [Nagoya Math. J. 59, 169--198 (1975; Zbl 0335.10031)].

11E45Analytic theory of forms
11R42Zeta functions and $L$-functions of global number fields
11R80Totally real fields
11R27Units and factorization
11F55Groups and their modular and automorphic forms (several variables)