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.