Vipul: Created page with '==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 $\...'$

2009-05-06T23:37:56Z

Created page with '==Definition== This is a generalization of the Riemann zeta-function to the ring of integers in a number field. Suppose <math>K</math> is a number field and <math>\...'

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==Definition==

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

Suppose <math>K</math> is a [[number field]] and <math>\mathcal{O}</math> is the [[ring of integers in a number field|ring of integers]] in <math>K</math>. Then, the '''Dedekind zeta-function''' for <math>K</math> is defined by the following series:

<math>\zeta_K(s) := \sum_I \frac{1}{(N(I))^s}</math>.

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

The series is absolutely convergent only for <math>\operatorname{Re}(s) > 1</math>, but it can be extended to a [[meromorphic function]] on the whole of <math>\mathbb{C}</math>, with a unique simple pole at <math>s = 1</math>.

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

Dedekind zeta-function - Revision history

Vipul: Created page with '==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 \...'

Vipul: Created page with '==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 $\...'$