Vipul: Created page with '==Statement== Suppose $K$ is a number field and $\mathcal{O}$ is the ring of integers in $K$ . The hyp...'

2009-05-07T01:34:25Z

Created page with '==Statement== Suppose <math>K</math> is a number field and <math>\mathcal{O}</math> is the ring of integers in <math>K</math>. The hyp...'

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==Statement==

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>. The hypothesis states the following: all the nontrivial zeros of the [[Dedekind zeta-function]] have real part <math>1/2</math>.

Here, the Dedekind zeta-function is defined as:

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

The sum is overall nonzero ideals of <math>\mathcal{O}</math>, and <math>N(I)</math> is the index of <math>I</math> in <math>\mathcal{O}</math> as an additive subgroup.

This result is sometimes termed the '''generalized Riemann hypothesis''' or the '''extended Riemann hypothesis''', but that name is typically used for the [[generalized Riemann hypothesis]] involving [[Dirichlet L-function]]s.

Generalization of Riemann hypothesis for number fields - Revision history

Vipul: Created page with '==Statement== Suppose K is a number field and \mathcal{O} is the ring of integers in K. The hyp...'

Vipul: Created page with '==Statement== Suppose $K$ is a number field and $\mathcal{O}$ is the ring of integers in $K$ . The hyp...'