�ber die Werte der Dedekindschen Zetafunktion

Klingen, Helmut

doi:10.1007/bf01451369

Cited by 75 publications

(44 citation statements)

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“…Since it is only (K(2) which can be interpreted geometrically in this way, we did not get a formula for (r(2m), m>l. However, we conjecture that an analogous result holds here, namely: (3) agrees with the function A(x) in Theorem 1.…”

Section: ~R(2)=~-x ~ Cva(x~l)a(xvs) (Finite Sum)mentioning

confidence: 49%

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Hyperbolic manifolds and special values of Dedekind zeta-functions

Zagier

1986

Invent Math

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Section: ~R(2)=~-x ~ Cva(x~l)a(xvs) (Finite Sum)mentioning

confidence: 49%

“…A famous theorem, proved by Euler in 1734, is that the sum rational multiple of ~2m for all natural numbers m: This result was generalized some years ago by Klingen [3] and Siegel [5], who showed that for an arbitrary totally real number field K the value of the Dedekind zeta function 1 …”

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confidence: 97%

Hyperbolic manifolds and special values of Dedekind zeta-functions

Zagier

1986

Invent Math

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“…Via C.L. Siegel and H. Klingen [4,8], if χ denotes a Dirichlet character and K is a totally real number field, then the Dirichlet L-function L(χ, s) lies in Q(χ) when s takes negative integer values. For a representation ρ, the associated character will simply be written as χ ρ .…”

Section: Introductionmentioning

confidence: 99%

Values of twisted Artin L-functions

Ward

2014

Arch. Math.

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ABSTRACT. This note gives a simple proof that certain values of Artin's L-function, for a representation ρ with character χ ρ , are stable under twisting by an even Dirichlet character χ, up to the dim(ρ)th power of the Gauss sum τ(χ) and an element generated over Q by the values of χ and χ ρ . This extends a result due to J. Coates and S. Lichtenbaum. INTRODUCTIONWe let K|F denote a finite Galois extension of algebraic number fields, and χ will denote the character associated with a representation of Gal(K|F). Via C.L. Siegel and H. Klingen [4,8], if χ denotes a Dirichlet character and K is a totally real number field, then the Dirichlet L-function L(χ, s) lies in Q(χ) when s takes negative integer values. For a representation ρ, the associated character will simply be written as χ ρ . For general representations ρ when K is a finite Galois extension of Q, J. Coates and S. Lichtenbaum [1] decomposed the factors at infinity in the functional equation [5, VII.12.6to show that, at a negative integer s = m that is a critical point for the Artin Lfunction L(K|Q, χ ρ , s), either (I) m is odd, and the fixed field K ρ of the kernel of ρ is totally real; or (II) K ρ is totally imaginary, conjugation is central in Gal(K ρ |Q), and χ ρ (σ ) = − dim(ρ). They then employed Brauer's theorem on induced characters and functorial properties of Artin's L-function to prove that L(K|Q, χ ρ , m) ∈ Q(χ ρ ) when m is a negative integer and critical point of L(K|Q, χ ρ , s) (for all other negative integers m, this L-function trivially takes the value zero).We consider a finite Galois extension K|Q, and a representation χ ρ of Gal(K|Q). We let χ denote an even Dirichlet character, which is viewed as either a onedimensional character of Gal(K|Q) acting trivially on the conjugation automorphism, or any one-dimensional character of Gal(K|Q) if K is totally real. We also let Q(χ ρ , χ) denote the field generated over Q by the values of χ ρ and χ. The 1991 Mathematics Subject Classification. 11F67, 11F80, 11L05, 11M06.

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“…Note that the summation in (1.8) is taken over totally positive elements of a −1 . Special values of ζ(a, f∞, s) at negative integers turns out to be rational numbers (see [Kli62], [Sie69] and [Shi76]). These rational numbers satisfy many remarkable congruence relations which have been exploited by many number theorists to construct various p-adic objects.…”

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confidence: 99%

Functional equation for partial zeta functions twisted by additive characters

Chapdelaine

2009

Acta Arith.

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�ber die Werte der Dedekindschen Zetafunktion

Cited by 75 publications

References 10 publications

Hyperbolic manifolds and special values of Dedekind zeta-functions

Hyperbolic manifolds and special values of Dedekind zeta-functions

Values of twisted Artin L-functions

Functional equation for partial zeta functions twisted by additive characters

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