# Dedekind zeta function

Let $K$ be a number field with ring of integers $\mathcal{O}_{K}$. Then the Dedekind zeta function of $K$ is the analytic continuation of the following series:

 $\zeta_{K}(s)=\sum_{I\subset\mathcal{O}_{K}}(N^{K}_{\mathbb{Q}}(I))^{-s}$

where $I$ ranges over non-zero ideals of $\mathcal{O}_{K}$, and $N^{K}_{\mathbb{Q}}(I)=|\mathcal{O}_{K}:I|$ is the norm of $I$.

This converges for $\Re(s)>1$, and has a meromorphic continuation to the whole plane, with a simple pole at $s=1$, and no others.

The Dedekind zeta function has an Euler product expansion,

 $\zeta_{K}(s)=\prod_{\mathfrak{p}}\frac{1}{1-(N^{K}_{\mathbb{Q}}(\mathfrak{p}))% ^{-s}}$

where $\mathfrak{p}$ ranges over prime ideals of $\mathcal{O}_{K}$. The Dedekind zeta function of $\mathbb{Q}$ is just the Riemann zeta function.

Title Dedekind zeta function DedekindZetaFunction 2013-03-22 13:20:23 2013-03-22 13:20:23 bwebste (988) bwebste (988) 9 bwebste (988) Definition msc 11M06 msc 11R42 RiemannZetaFunction ClassNumberFormula