Hyperbolic manifolds and special values of Dedekind zeta-functions
type: Item
instance of: scholarly article
title: Hyperbolic manifolds and special values of Dedekind zeta-functions
zbMATH Open document ID: 0591.12014
DOI: 10.1007/BF01388964
author: Don Zagier
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143313
review text: 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 [\textit{C. L. Siegel}, Nachr. Akad. Wiss. Göttingen, II. Math.- Phys. Kl. 1969, 87-102 (1969; Zbl 0186.088); and \textit{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.
reviewed by: Walter D. Neumann
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3948420
zbMATH Keywords: Dedekind zeta-function
zbMATH Keywords: volumes of hyperbolic manifolds
zbMATH Keywords: Lobachevskij
zbMATH Keywords: dilogarithm
MaRDI profile type: Publication
cites work: Q3927897
cites work: Über die Werte der Dedekindschen Zetafunktion
cites work: Q3916824
cites work: Q5579662
cites work: Hyperbolic structures on 3-manifolds. I: Deformation of acylindrical manifolds
cites work: Arithmétique des algèbres de quaternions
description: scientific article; zbMATH DE number 3948420