## Hyperbolic manifolds and special values of Dedekind zeta-functions.(English)Zbl 0591.12014

Let K be any number field of discriminant D with r real places and s complex places. Let $$\zeta_ K$$ be the associated Dedekind zeta- function. In this paper the author conjectures, and proves in several cases, that the value of $$\zeta_ K(2m)$$ is $$\pi^{2m(r+s)}/ \sqrt{| D|}$$ times a rational linear combination of products of s values of $$A_ m(x)$$ at algebraic arguments. Here $$A_ m(x)$$ is the real-valued function $A_ m(x)=\frac{2^{2m-1}}{(2m- 1)!}\int^{\infty}_{0}\frac{t^{2m-1} dt}\quad {x \sinh^ 2t+x^{- 1} \cosh^ 2t}.$ The special case $$s=0$$ is a result of Siegel and Klingen [C. L. Siegel, Nachr. Akad. Wiss. Göttingen, II. Math.- Phys. Kl. 1969, 87-102 (1969; Zbl 0186.088); and H. Klingen, Math. Ann. 145, 265-272 (1962; Zbl 0101.030)], generalizing Euler’s famous theorem of 1734 that the value of the Riemann zeta-function at an even argument 2m is a rational multiple of $$\pi^{2m}.$$
Using geometric arguments involving the volumes of hyperbolic manifolds, the author proves the conjecture for $$m=1$$ and arbitrary K (Theorem 1; in this case $$A(x)=A_ 1(x)$$ can be simply described in terms of the Lobachevskij function and hence also in terms of the classical dilogarithm function). The necessary algebraic arguments of A(x) can always be chosen in quite specific field extensions of degree at most 8 over K, and even in $${\mathbb{Q}}(\sqrt{| D|})$$ in the imaginary quadratic case.
Using ”routine number-theoretic tools”, the author also proves the conjecture for K abelian over $${\mathbb{Q}}$$ (Theorem 2). In the overlap of Theorems 1 and 2, Theorem 1 gives a stronger statement, since the formulae resulting from the number-theoretic approach do not yield arguments of bounded degree over K. One also obtains many nontrivial relations between values of A(x) at algebraic arguments. Numerical examples and computational methods are also described.
Reviewer: W.D.Neumann

### MSC:

 11R42 Zeta functions and $$L$$-functions of number fields 53C30 Differential geometry of homogeneous manifolds 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$

### Citations:

Zbl 0186.088; Zbl 0101.030
Full Text:

### References:

  Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Super. Pisa8, 1-33 (1981) (=Oeuvres3, 617-649) · Zbl 0473.57003  Hörsch, K.-H.: Ein Verfahren zur Berechnung vonL-Reihen. Diplomarbeit, Bonn 1982  Klingen, H.: Über die Werte der Dedekindschen Zetafunktionen. Math. Ann.145, 265-272 (1962) · Zbl 0101.03002  Lewin, L.: Polylogarithms and Associated Functions. New York-Oxford: North Holland 1981 · Zbl 0465.33001  Siegel, C.L.: Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Gött.10, 87-102 (1969) (=Ges. Abh.4, 82-97) · Zbl 0186.08804  Thurston, W.P.: The Geometry and Topology of Three-Manifolds (Chapter 7, ?Computation of Volume?, by J. Milnor). Mimeographed lecture notes, Princeton University 1979  Thurston, W.P.: Hyperbolic structures on 3-manifolds, I. In: Deformation of acylindric manifolds. Ann. Math. (to appear) · Zbl 0668.57015  Vignéras, M.-F.: Arithmétique des Algèbres de Quaternions. Lect. Notes No. 800. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0422.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.