# Generalization of Riemann hypothesis for number fields

From Number

## Statement

Suppose is a number field and is the ring of integers in . The hypothesis states the following: all the nontrivial zeros of the Dedekind zeta-function have real part .

Here, the Dedekind zeta-function is defined as:

.

The sum is overall nonzero ideals of , and is the index of in 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-functions.