PlanetMath: Dedekind zeta function

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Dedekind zeta function

(Definition)

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

$\displaystyle \zeta_K(s)=\sum_{I\subset\mathcal{O}_K} (N^K_{\mathbb{Q}}(I))^{-s}$

where

ranges over non-zero ideals of $\mathcal{O}_K$ , and $N^K_{\mathbb{Q}}(I)=\vert\O _K:I\vert$ is the norm of

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

The Dedekind zeta function has an Euler product expansion,

$\displaystyle \zeta_K(s)= \prod_{\mathfrak{p}}\frac{1}{1-(N^K_{\mathbb{Q}}(\mathfrak{p}))^{-s}}$

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

"Dedekind zeta function" is owned by bwebste.

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Attachments:: generalized Riemann hypothesis (Definition) by mathcam
Siegel-Klingen Theorem (Theorem) by alozano
Factorization of the Dedekind zeta function of an abelian number field (Theorem) by alozano
values of Dedekind zeta functions of real quadratic number fields at negative integers (Application) by alozano

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AMS MSC:	11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
	11R42 (Number theory :: Algebraic number theory: global fields :: Zeta functions and $L$-functions of number fields)

Pending Errata and Addenda

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