Let be a totally real algebraic number field of degree . If and are mutually prime integrals of , let , where the summation is over all integral ideals of which are in the same narrow ray class mod as . 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 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 -dimensional generalization of the classical Hankel type integral for the zeta-function. An essential point in the argument is the partition of into a disjoint union of “open simplicial cones”, i.e. subsets of the form for any set of linearly independent vectors . 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 be a totally imaginary quadratic extension of and let and denote the class numbers of and , respectively. The author derives a formula for which may be regarded as an affirmative answer to the Hecke conjecture that 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)].