Dedekind zeta function

From specialfunctionswiki
Jump to: navigation, search

Let $F$ be a a finite field extension of the rational numbers. The Dedekind zeta function of $F$ is $$\zeta_F(z)=\displaystyle\sum_{\mathfrak{a}} N(\mathfrak{a})^{-z}; \mathrm{Re}(z)>1,$$ where the sum is over the nontrivial ideals $\mathfrak{a}$ of the ring of integers $\mathcal{O}_F$ of $F$ and $N(\mathfrak{a})$ denotes the norm of the ideal $\mathfrak{a}$.

Properties[edit]

Theorem: The following Euler product holds: $$\zeta_F(z)=\displaystyle\prod_{\mathfrak{p}} \dfrac{1}{1-N(\mathfrak{p})^{-z}},$$ where $\mathfrak{p}$ denotes a nontrivial prime ideal of $\mathcal{O}_F$.

Proof:

References[edit]

Panorama of zeta functions