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

Zagier, Don

Invent. Math. 83, 285-301 (1986).

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 \(L(s, \chi)\)

Keywords:

Dedekind zeta-function; volumes of hyperbolic manifolds; Lobachevskij; dilogarithm

Citations:

Zbl 0186.088; Zbl 0101.030

References:

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