Editing Chebotarev density theorem (section)

==Statement==
Let ''L'' be a finite Galois extension of a number field ''K'' with Galois group ''G''. Let ''X'' be a subset of ''G'' that is stable under conjugation. The set of primes ''v'' of ''K'' that are unramified in ''L'' and whose associated Frobenius conjugacy class ''F''<sub>v</sub> is contained in ''X'' has density
:<math>\frac{\#X}{\#G}.</math><ref name="Section">Section I.2.2 of Serre</ref>
The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.<ref>{{cite web |url= http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf|title=The Chebotarev Density Theorem |last=Lenstra |first=Hendrik |date=2006 |access-date=7 June 2018 }}</ref>

===Effective version===
The [[Generalized Riemann hypothesis]] implies an [[Effective results in number theory|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 unramified primes of ''K'' of norm below ''x'' with Frobenius conjugacy class in ''C'' is
:<math>\frac{|C|}{|G|}\Bigl(\mathrm{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.

The effective form of the Chebotarev density theory becomes much weaker without GRH. Take ''L'' to be a finite Galois extension of ''Q'' with Galois group ''G'' and degree ''d''. Take <math>\rho</math> to be a nontrivial irreducible representation of ''G'' of degree ''n'', and take <math>\mathfrak{f}(\rho)</math> to be the Artin conductor of this representation. Suppose that, for <math>\rho_0</math> a subrepresentation of <math>\rho \otimes \rho</math> or <math> \rho \otimes \bar{\rho}</math>, <math>L(\rho_0, s)</math> is entire; that is, the Artin conjecture is satisfied for all <math>\rho_0</math>. Take <math>\chi_{\rho}</math> to be the character associated to <math>\rho</math>. Then there is an absolute positive <math>c</math> such that, for <math> x \ge 2</math>,
:<math>\sum_{p \le x, p \not\mid \mathfrak{f}(\rho)} \chi_{\rho}(\text{Fr}_p) \log p = rx + O\biggl(\frac{x^{\beta}}{\beta} + x\exp\biggl(\frac{-c(dn)^{-4} \log x }{3\log \mathfrak{f}(\rho) + \sqrt{\log x}}\biggr) (dn \log (x\mathfrak{f}(\rho))\biggr),</math>
where <math>r</math> is 1 if <math>\rho</math> is trivial and is otherwise 0, and where <math>\beta</math> is an [[Siegel zero|exceptional real zero]] of <math>L(\rho, s)</math>; if there is no such zero, the <math>x^{\beta}/\beta</math> term can be ignored. The implicit constant of this  expression is absolute. <ref>{{cite book | last1=Iwaniec | first1=Henryk | last2=Kowalski | first2=Emmanuel| title=Analytic Number Theory| year= 2004| location=Providence, RI| publisher=American Mathematical Society| page=111}}</ref>

===Infinite extensions===
The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension ''L'' / ''K'' that is unramified outside a finite set ''S'' of primes of ''K'' (i.e. if there is a finite set ''S'' of primes of ''K'' such that any prime of ''K'' not in ''S'' is unramified in the extension ''L'' / ''K''). In this case, the Galois group ''G'' of ''L'' / ''K'' is a [[profinite group]] equipped with the Krull topology. Since ''G'' is compact in this topology, there is a unique [[Haar measure]] μ on ''G''. For every prime ''v'' of ''K'' not in ''S'' there is an associated Frobenius conjugacy class ''F''<sub>v</sub>. The Chebotarev density theorem in this situation can be stated as follows:<ref name="Section" />

:Let ''X'' be a subset of ''G'' that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes ''v'' of ''K'' not in ''S'' such that ''F''<sub>v</sub> ⊆ X has density
::<math>\frac{\mu(X)}{\mu(G)}.</math>

This reduces to the finite case when ''L'' / ''K'' is finite (the Haar measure is then just the counting measure).

A consequence of this version of the theorem is that the Frobenius elements of the unramified primes of ''L'' are dense in ''G''.