Editing Analytic number theory (section)

{{Short description|Exploring properties of the integers with complex analysis}}
[[Image:Complex zeta.jpg|right|thumb|300px|Riemann zeta function ''ζ''(''s'') in the [[complex plane]]. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while [[hue]] encodes the value's [[Argument (complex analysis)|argument]].]]

In [[mathematics]], '''analytic number theory''' is a branch of [[number theory]] that uses methods from [[mathematical analysis]] to solve problems about the [[integer]]s.{{sfn|Apostol|1976|p=7|ignore-err=yes}} It is often said to have begun with [[Peter Gustav Lejeune Dirichlet]]'s 1837 introduction of [[Dirichlet L-function|Dirichlet ''L''-function]]s to give the first proof of [[Dirichlet's theorem on arithmetic progressions]].{{sfn|Apostol|1976|p=7|ignore-err=yes}}{{sfn|Davenport|2000|p=1}} It is well known for its results on [[prime numbers]] (involving the [[Prime Number Theorem]] and [[Riemann zeta function]]) and [[additive number theory]] (such as the [[Goldbach conjecture]] and [[Waring's problem]]).