Generalization of Riemann hypothesis for number fields

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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.