Dedekind zeta function
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In mathematics, the Dedekind zeta-function is a Dirichlet series defined for any algebraic number field K, and denoted ζK(s) where s is a complex variable. It is the infinite sum
taken over all ideals I of the ring of integers OK of K, with 
. Here NI denotes the norm of I (to the rational field Q): it is equal to the cardinality of
- OK / I,
in other words, the number of residue classes modulo I. In the case K = Q this definition reduces to the Riemann zeta function.
The properties of ζK(s) as a meromorphic function turn out to be of considerable significance in algebraic number theory. It has an Euler product, with a factor for a given prime number p the product over all the prime ideals P of OK dividing p of
This is the expression in analytic terms of the uniqueness of prime factorization of the ideals I.
It is known (proved first in general by Erich Hecke) that ζK(s) does have an analytic continuation to the whole complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is an important quantity, involving invariants of the unit group and class group of K; details are at class number formula. There is a functional equation for the Dedekind zeta-function, relating its values at s and 1 − s.
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 an L-function
- L(s,χ)
where χ is a Jacobi symbol 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 group G, its Dedekind zeta-function has a comparable factorization in terms of Artin L-functions. These are attached to linear representations of G.
