Editing Analytic number theory (section)

===Dirichlet===
{{main|Johann Peter Gustav Lejeune Dirichlet}}
[[Johann Peter Gustav Lejeune Dirichlet]] is credited with the creation of analytic number theory,<ref name=Princeton>{{cite book| last = Gowers| first = Timothy |author1-link=Timothy Gowers|author2-link = June Barrow-Green| author3-link = Imre Leader|author2=June Barrow-Green |author3=Imre Leader | title=The Princeton companion to mathematics| url = https://archive.org/details/princetoncompanio00gowe| year=2008| publisher=Princeton University Press| isbn= 978-0-691-11880-2| pages= 764–765}}</ref> a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published [[Dirichlet's theorem on arithmetic progressions]], using [[mathematical analysis]] concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the [[Dirichlet character]]s and [[Dirichlet L-function|L-functions]].<ref name=Princeton/><ref name=Kanemitsu>{{cite book| last = Kanemitsu| first = Shigeru|author2=Chaohua Jia| title=Number theoretic methods: future trends | year=2002| publisher=Springer| isbn= 978-1-4020-1080-4| pages= 271–274}}</ref> In 1841 he generalized his arithmetic progressions theorem from integers to the [[Ring (mathematics)|ring]] of [[Gaussian integer]]s <math>\mathbb{Z}[i]</math>.<ref name=Elstrodt>{{cite journal | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | access-date = 2007-12-25 | archive-date = 2008-03-07 | archive-url = https://web.archive.org/web/20080307174514/http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | url-status = dead }}</ref>