Dedekind zeta-function

Definition

This is a generalization of the Riemann zeta-function to the ring of integers in a number field.

Suppose $K$ is a number field and $\mathcal{O}$ is the ring of integers in $K$. Then, the Dedekind zeta-function for $K$ is defined by the following series:

${\zeta }_K(s):={\sum }_I\frac{1}{(N(I){)}^s}$.

Here, the sum is over all nonzero ideals $I$ of $\mathcal{O}$, and $N(I)$ denotes the norm of the ideal $I$, which is equal to the index of $I$ as a subgroup of $\mathcal{O}$.

The series is absolutely convergent only for $\mathrm{R}\mathrm{e}(s)>1$, but it can be extended to a meromorphic function on the whole of $\mathbb{C}$, with a unique simple pole at $s=1$.

Note that when $K=\mathbb{Q}$, $\mathcal{O}=\mathbb{Z}$, and the Dedekind zeta-function equals the Riemann zeta-function.