Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numberss with real part Re > 1 by the Dirichlet series where I ranges through the non-zero ideals of the ring of integersOK of K and NK/Q denotes the absolute norm of I. This sum converges absolutely for all complex numbers s with real part Re > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.
Euler product
The Dedekind zeta function of K has an Euler product which is a product over all the prime idealsP of OK This is the expression in analytic terms of the uniqueness of prime factorization of the idealsI in OK. For Re > 1, ζK is non-zero.
first proved that ζK has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K. The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote the discriminant of K, let r1 denote the number of real places of K, and let and where Γ is the Gamma function. Then, the functions satisfy the functional equation
Special values
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode important arithmetic data of the fieldK. For example, the analytic class number formula relates the residue at s = 1 to the class numberh of K, the regulatorR of K, the number w 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 OK and the leading term is given by It follows from the functional equation that. Combining the functional equation and the fact that Γ is infinite at all integers less than or equal to zero yields that ζK vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real. In the totally real case, Carl Ludwig Siegel showed that ζK is a non-zero rational number at negative odd integers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K.
Relations to other ''L''-functions
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. For example, when K is a quadratic field this shows that the ratio is the L-function L, where χ is a Jacobi symbol used as Dirichlet character. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss. In general, if K is a Galois extension of Q with Galois groupG, 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. The relation with Artin L-functions shows that if L/K is a Galois extension then is holomorphic : for general extensions the result would follow from the Artin conjecture for L-functions. Additionally, ζK is the Hasse–Weil zeta function of SpecOK and the motivic L-function of the motive coming from the cohomology of Spec K.
Arithmetically equivalent fields
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. used Gassmann triples to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. 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.
