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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 $𝐛$ and $𝐟$ are mutually prime integrals of $F$, let $\zeta \left(𝐛,𝐟,s\right)=\sum N{\left(𝐚\right)}^{-s}$, where the summation is over all integral ideals $𝐚$ of $F$ which are in the same narrow ray class mod $𝐟$ as $𝐛$. 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 $\zeta \left(𝐛,𝐟,s\right)$ 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 ${ℝ}^{n}$ into a disjoint union of “open simplicial cones”, i.e. subsets of the form $\left\{{x}_{1}{v}_{1}+\cdots +{x}_{r}{v}_{r}\mid {x}_{i}>0\right\}$ for any set of linearly independent vectors ${v}_{1},\cdots ,{v}_{r}$. 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)].

##### MSC:
 11E45 Analytic theory of forms 11R42 Zeta functions and $L$-functions of global number fields 11R80 Totally real fields 11R27 Units and factorization 11F55 Groups and their modular and automorphic forms (several variables)