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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 ζ(𝐛,𝐟,s)=N(𝐚) -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 ζ(𝐛,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 n into a disjoint union of “open simplicial cones”, i.e. subsets of the form {x 1 v 1 ++x r v r x i >0} for any set of linearly independent vectors v 1 ,,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:
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)