where ranges through the non-zero ideals of the ring of integers of . Here denotes the norm of (to the rational field ). It is equal to the cardinality of , in other words, the number of residue classes modulo . This sum converges absolutely for all complex numbers with real part . In the case this definition reduces to the Riemann zeta function.
This is the expression in analytic terms of the uniqueness of prime factorization of the ideals .
It is known (proved first in general by Erich Hecke) that 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 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.
Researchers from University of Lethbridge Report Details of New Studies and Findings in the Area of Number Theory.
Mar 06, 2012; "Let K be a number field, n(K) be its degree, and d(K) be the absolute value of its discriminant," scientists writing in the...