Editing Riemann zeta function (section)

{{Short description|Analytic function in mathematics}}

[[File:Cplot zeta.svg|right|thumb|250px|The Riemann zeta function {{math|''ζ''(''z'')}} plotted with [[domain coloring]].<ref>{{cite web|url=http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb |title=Jupyter Notebook Viewer|website=Nbviewer.ipython.org|access-date=2017-01-04}}</ref>]]
[[File:Riemann-Zeta-Detail.png|right|thumb|200px|The pole at <math>z=1</math> and two zeros on the critical line.]]
The '''Riemann zeta function''' or '''Euler–Riemann zeta function''', denoted by the [[Greek alphabet|Greek letter]] {{math|''ζ''}} ([[zeta]]), is a [[function (mathematics)|mathematical function]] of a [[complex variable]] defined as <math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math> for {{nowrap|<math>\operatorname{Re}(s) > 1</math>,}} and its [[analytic continuation]] elsewhere.<ref name=":0" />

The Riemann zeta function plays a pivotal role in [[analytic number theory]] and has applications in [[physics]], [[probability theory]], and applied [[statistics]].

[[Leonhard Euler]] first introduced and studied the function over the [[real numbers|reals]] in the first half of the eighteenth century. [[Bernhard Riemann]]'s 1859 article "[[On the Number of Primes Less Than a Given Magnitude]]" extended the Euler definition to a [[complex number|complex]] variable, proved its [[meromorphic]] continuation and [[functional equation]], and established a relation between its [[Root of a function|zeros]] and [[prime number theorem|the distribution of prime numbers]].  This paper also contained the [[Riemann hypothesis]], a [[conjecture]] about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in [[pure mathematics]].<ref>{{cite web | last=Bombieri | first=Enrico | url=http://www.claymath.org/sites/default/files/official_problem_description.pdf | title=The Riemann Hypothesis – official problem description | publisher=[[Clay Mathematics Institute]] | access-date=2014-08-08 | archive-date=22 December 2015 | archive-url=https://web.archive.org/web/20151222090027/http://www.claymath.org/sites/default/files/official_problem_description.pdf | url-status=dead }}</ref>

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, {{math|''ζ''(2)}}, provides a solution to the [[Basel problem]]. In 1979 [[Roger Apéry]] proved the irrationality of {{math|[[Apéry's constant|''ζ''(3)]]}}. The values at negative integer points, also found by Euler, are [[rational number]]s and play an important role in the theory of [[modular form]]s. Many generalizations of the Riemann zeta function, such as [[Dirichlet series]], [[Dirichlet L-function|Dirichlet {{mvar|L}}-functions]] and [[L-function|{{mvar|L}}-functions]], are known.