Editing Generalized Riemann hypothesis (section)

== Extended Riemann hypothesis (ERH) ==

Suppose ''K'' is a [[number field]] (a finite-dimensional [[field extension]] of the [[rational number|rationals]] <math>\mathbb Q</math>) with [[algebraic number|ring of integers]] O<sub>''K''</sub> (this ring is the [[integral closure]] of the [[integer]]s <math>\mathbb Z</math> in ''K''). If ''a'' is an [[ring ideal|ideal]] of O<sub>''K''</sub>, other than the zero ideal, we denote its [[ideal norm|norm]] by ''Na''. The [[Dedekind zeta-function]] of ''K'' is then defined by
:<math>
\zeta_K(s) = \sum_a \frac{1}{(Na)^s}
</math>
for every complex number ''s'' with real part >&nbsp;1. The sum extends over all non-zero ideals ''a'' of O<sub>''K''</sub>.

The Dedekind zeta-function satisfies a functional equation and can be extended by [[analytic continuation]] to the whole complex plane. The resulting function encodes important information about the number field ''K''. The extended Riemann hypothesis asserts that for every number field ''K'' and every complex number ''s'' with ζ<sub>''K''</sub>(''s'') = 0: if the real part of ''s'' is between 0 and 1, then it is in fact 1/2.

The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be <math>\mathbb Q</math>, with ring of integers <math>\mathbb Z</math>.

The ERH implies an effective version<ref>{{cite journal|first1=J.C.|last1=Lagarias|first2=A.M.|last2=Odlyzko|title=Effective Versions of the Chebotarev Theorem|journal=Algebraic Number Fields|year=1977|pages=409–464}}</ref> of the [[Chebotarev density theorem]]: if ''L''/''K'' is a finite Galois extension with Galois group ''G'', and ''C'' a union of conjugacy classes of ''G'', the number of [[Ramification (mathematics)#In algebraic number theory|unramified primes]] of ''K'' of norm below ''x'' with Frobenius conjugacy class in ''C'' is
:<math>\frac{|C|}{|G|}\Bigl(\operatorname{Li}(x)+O\bigl(\sqrt x(n\log x+\log|\Delta|)\bigr)\Bigr),</math>
where the constant implied in the big-O notation is absolute, ''n'' is the degree of ''L'' over ''Q'', and Δ its discriminant.