Sunada noticed that the method of constructing number fields with the same Dedekind zeta function could be adapted to compact manifolds.

* Dedekind zeta function

Just the same shape of functional equation holds for the Dedekind zeta function of a number field K, with an appropriate gamma-factor that depends only on the embeddings of K ( in algebraic terms, on the tensor product of K with the real field ).

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ < sub > K </ sub >( s ), is a generalization of the Riemann zeta function — which is obtained by specializing to the case where K is the rational numbers Q.

The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to P. G. L.

Let K be an algebraic number field K. Its Dedekind zeta function is first defined for complex numbers s with real part Re ( s ) > 1 by the Dirichlet series

The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let Δ < sub > K </ sub > denote discriminant of K, let r < sub > 1 </ sub > ( resp.

Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode ( at least conjecturally ) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h ( K ) of K, the regulator R ( K ) of K, the number w ( K ) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of O < sub > K </ sub > and the leading term is given by

For the case in which K is an abelian extension of Q, its Dedekind zeta function can be written as a product of Dirichlet L-functions.

In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G.

Two fields are called arithmetically equivalent if they have the same Dedekind zeta function.

In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.

ca: Funció zeta de Dedekind

es: Función zeta de Dedekind

pt: Função zeta de Dedekind

In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions using integration over ideles and duality of adeles ; this method was also independently obtained by John Tate and it is sometimes called Tate's thesis or the Iwasawa-Tate theory.

More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke L-series.

* Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function.

* Dedekind psi function

In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain which is a subring of the rational function field of an elliptic curve, and conjectured that such an " elliptic " construction should be possible for a general abelian group ( Rosen 1976 ).

* The modular discriminant Δ ( τ ) is proportional to the 24th power of the Dedekind eta function η ( τ ): Δ ( τ ) = ( 2π )< sup > 12 </ sup > η ( τ )< sup > 24 </ sup >.

* The Dedekind eta function η ( τ ), a modular form

* Dedekind eta function ( math ), a complex function defined in the upper plane

The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive.

Note that where is the Dedekind eta function.

Let η be the Dedekind eta function.

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η ( τ ), a modular function involving a 24th root, and which explains the 24 in the approximation.

In mathematics, the Chowla – Selberg formula is the evaluation of a certain product of values of the Gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers.

The function η is the Dedekind eta function, and h is the class number, and w is the number of roots of unity.

which represents ( up to a normalizing constant ) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve ; and the 24-th power of the Dedekind eta function.

The ring of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one ( to see the last property, observe that for any nonzero ideal I of R, R / I is finite and recall that a finite integral domain is a field ), so by ( DD4 ) R is a Dedekind domain.

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the integral closure S of R in L to be a Dedekind domain, but it is not guaranteed.

The Dedekind zeta-function of K is then defined by

In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

Let A be a Dedekind domain with the field of fractions K and B be the integral closure of A in a finite separable extension L of K. ( In particular, B is Dedekind then.

More generally, the PBW theorem as formulated above extends to cases such as where ( 1 ) L is a flat K-module, ( 2 ) L is torsion-free as an abelian group, ( 3 ) L is a direct sum of cyclic modules ( or all its localizations at prime ideals of K have this property ), or ( 4 ) K is a Dedekind domain.

# There is some Dedekind valuation ν on the field of fractions K of R, such that R =

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

William Everdell, for example, has argued that modernism began in the 1870s, when metaphorical ( or ontological ) continuity began to yield to the discrete with mathematician Richard Dedekind's ( 1831 – 1916 ) Dedekind cut, and Ludwig Boltzmann's ( 1844 – 1906 ) statistical thermodynamics.

Over an arbitrary Dedekind domain one has

If the ring has special properties, this hierarchy may collapse, i. e. for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.

In mathematical logic, the phrase Cantor – Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry.

Since G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function ( for the trivial representation ) and an L-function of Dirichlet's type for the signature representation.