Dedekind zeta function

Let K be a number field with ring of integers π’ͺK. Then the Dedekind zeta function of K is the analytic continuation of the following series:


where I ranges over non-zero ideals of π’ͺK, and Nβ„šK(I)=|π’ͺK:I| is the norm of I.

This converges for β„œβ‘(s)>1, and has a meromorphic continuation to the whole plane, with a simple poleMathworldPlanetmathPlanetmath at s=1, and no others.

The Dedekind zeta function has an Euler productMathworldPlanetmath expansion,


where 𝔭 ranges over prime idealsMathworldPlanetmath of π’ͺK. The Dedekind zeta function of β„š is just the Riemann zeta functionDlmfDlmfMathworldPlanetmath.

Title Dedekind zeta function
Canonical name DedekindZetaFunction
Date of creation 2013-03-22 13:20:23
Last modified on 2013-03-22 13:20:23
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 9
Author bwebste (988)
Entry type Definition
Classification msc 11M06
Classification msc 11R42
Related topic RiemannZetaFunction
Related topic ClassNumberFormula