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 $\operatorname {Re} (s)>1$, 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.