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{| border="1" style="float: right; border-collapse: collapse; margin: 0 0 0 1em;"
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| {{gaps|1.00110|01110|11101...}}
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| [[Decimal]]
| {{gaps|1.20205|69031|59594|2854...}}
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| [[Hexadecimal]]
| {{gaps|1.33BA0|04F00|621383...}}
|-
| [[Continued fraction]]
| <math>1 + \frac{1}{4 + \cfrac{1}{1 + \cfrac{1}{18 + \cfrac{1}{\ddots\qquad{}}}}}</math><br><small>Note that this continuing fraction is not [[Periodic continued fraction|periodic]].</small>
|}


In [[mathematics]], '''Apéry's constant''' is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's [[gyromagnetic ratio]] using quantum electrodynamics. It also arises in conjunction with the [[gamma function]] when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the [[Debye model]] and the [[Stefan–Boltzmann law]].  
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It is defined as the number ζ(3),
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:<math>\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}=1+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \frac{1}{6^3} + \frac{1}{7^3} + \frac{1}{8^3} + \frac{1}{9^3} + \cdots\,\!</math>
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where ζ is the [[Riemann zeta function]]. It has an approximate value of {{harv|Wedeniwski|2001}}
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:&zeta;(3) = {{gaps|1.20205|69031|59594|28539|97381|61511|44999|07649|86292...}} &nbsp; {{OEIS|id=A002117}}.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The [[reciprocal (mathematics)|reciprocal]] of this [[constant (mathematics)|constant]] is the [[probability]] that any three [[positive integer]]s, chosen at random, will be [[relatively prime]] (in the sense that as ''N ''goes to infinity, the probability that three positive integers less than ''N'' chosen uniformly at random will be relatively prime approaches this value).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Apéry's theorem==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
{{main|Apéry's theorem}}


This value was named for [[Roger Apéry]] (1916–1994), who in 1978 proved it to be [[irrational number|irrational]]. This result is known as ''[[Apéry's theorem]]''. The original proof is complex and hard to grasp, and shorter proofs have been found later, using [[Legendre polynomials]].  It is not known whether Apéry's constant is [[transcendental number|transcendental]].
==Demos==


Work by [[Wadim Zudilin]] and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2''n''+1) must be irrational,<ref>{{Citation |author=T. Rivoal |title=La fonction zeta de Riemann prend une infnité de valuers irrationnelles aux entiers impairs |journal=Comptes Rendus de l'Académie des Sciences. Série I. Mathématique |volume=331 |year=2000 |pages=267–270 |postscript=.}}</ref> and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.<ref>{{Citation |author=W. Zudilin |title=One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational |journal=Russ. Math. Surv. |year=2001 |volume=56 |pages=774–776 |doi=10.1070/RM2001v056n04ABEH000427 |postscript=. |issue=4}}</ref>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==Series representation==
In 1772, [[Leonhard Euler]] {{harv|Euler|1773}} gave the series representation {{harv|Srivastava|2000|loc=p. 571 (1.11)}}:


:<math>\zeta(3)=\frac{\pi^2}{7}
* accessibility:
\left[ 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {(2k+1)(2k+2) 2^{2k}} \right]</math>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
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** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


which was subsequently rediscovered several times.
==Test pages ==


[[Ramanujan]] gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include<ref>Bruce C. Berndt, ''Ramanujan's notebooks, Part II'' (1989), Springer-Verlag. ''See chapter 14, formulas 25.1 and 25.3''</ref>:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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:<math>\zeta(3)=\frac{7}{180}\pi^3 -2
*[[Inputtypes|Inputtypes (private Wikis only)]]
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}</math>
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
[[Simon Plouffe]] has developed other series{{harv|Plouffe|1998}}:
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
:<math>\zeta(3)= 14
\sum_{k=1}^\infty \frac{1}{k^3 \sinh(\pi k)}
-\frac{11}{2}
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}
-\frac{7}{2}
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} +1)}.
</math>
 
Similar relations for the values of <math>\zeta(2n+1)</math> are given in the article [[zeta constants]].
 
Many additional series representations have been found, including:
 
:<math>\zeta(3) = \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k+1)^3}</math>
 
:<math>\zeta(3) = \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^3}</math>
 
:<math>\zeta(3) = \frac{5}{2} \sum_{k=1}^\infty (-1)^{k-1} \frac{(k!)^2}{k^3 (2k)!}</math>
 
:<math>\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1}
\frac{56k^2-32k+5}{(2k-1)^2} \frac{((k-1)!)^3}{(3k)!}</math>
 
:<math>\zeta(3)=\frac{8}{7}-\frac{8}{7}\sum_{k=1}^\infty \frac{{\left( -1 \right) }^k\,2^{-5 + 12\,k}\,k\,
    \left( -3 + 9\,k + 148\,k^2 - 432\,k^3 - 2688\,k^4 + 7168\,k^5 \right) \,
    {k!}^3\,{\left( -1 + 2\,k \right) !}^6}{{\left( -1 + 2\,k \right) }^3\,
    \left( 3\,k \right) !\,{\left( 1 + 4\,k \right) !}^3}</math>
 
:<math>\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{205k^2 + 250k + 77}{64} \frac{(k!)^{10}}{((2k+1)!)^5}</math>
 
and
 
:<math>\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{P(k)}{24}
\frac{((2k+1)!(2k)!k!)^3}{(3k+2)!((4k+3)!)^3}</math>
 
where
 
:<math>P(k) = 126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463.\,</math>
 
Some of these have been used to calculate Apéry's constant with several million digits.
 
{{harvtxt|Broadhurst|1998}} gives a series representation that allows arbitrary [[binary digit]]s to be computed, and thus, for the constant to be obtained in nearly [[linear time]], and [[logarithmic space]].
 
==Other formulas==
Apéry's constant can be expressed in terms of the second-order [[polygamma function]] as
 
:<math>\zeta(3) = -\frac{1}{2} \, \psi^{(2)}(1).</math>
 
It can be expressed as the non-periodic  [[continued fraction]] [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] {{OEIS|id=A013631}}.
 
==Known digits==
 
The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.
 
{| class="wikitable" style="margin: 1em auto 1em auto"
|+ '''Number of known decimal digits of Apéry's constant ζ(3)'''
! Date || Decimal digits || Computation performed by
|-
| 1735 || 16 || [[Leonhard Euler]]
|-
| unknown || 16 || [[Adrien-Marie Legendre]]
|-
| 1887 || 32 || [[Thomas Joannes Stieltjes]]
|-
| 1996 || 520,000 || Greg J. Fee & [[Simon Plouffe]]
|-
| 1997 || 1,000,000 || Bruno Haible & Thomas Papanikolaou
|-
| May 1997 || 10,536,006 || Patrick Demichel
|-
| February 1998 || 14,000,074 || Sebastian Wedeniwski
|-
| March 1998 || 32,000,213 || Sebastian Wedeniwski
|-
| July 1998 || 64,000,091 || Sebastian Wedeniwski
|-
| December 1998 || 128,000,026 || Sebastian Wedeniwski {{harv|Wedeniwski|2001}}
|-
| September 2001 || 200,001,000 || Shigeru Kondo & Xavier Gourdon
|-
| February 2002 || 600,001,000 || Shigeru Kondo & Xavier Gourdon
|-
| February 2003 || 1,000,000,000 || Patrick Demichel & Xavier Gourdon
|-
| April 2006 || 10,000,000,000 || Shigeru Kondo & Steve Pagliarulo (see {{harvtxt|Gourdon|Sebah|2003}})
|-
| January 2009 || 15,510,000,000 || Alexander J. Yee & Raymond Chan (see {{harvtxt|Yee|Chan|2009}})
|-
| March 2009 || 31,026,000,000 || Alexander J. Yee & Raymond Chan (see {{harvtxt|Yee|Chan|2009}})
|-
| September 2010 || 100,000,001,000 || Alexander J. Yee (see [http://www.numberworld.org/digits/Zeta%283%29/ Yee])
|}
 
==Notes==
{{Reflist}}
 
==References==
*{{citation
| last = Euler
| first = Leonhard
| authorlink = Leonhard Euler
| year = 1773
| title = Exercitationes analyticae
| journal = Novi Commentarii academiae scientiarum Petropolitanae
| volume = 17
| pages = 173–204
| url = http://math.dartmouth.edu/~euler/docs/originals/E432.pdf
| language = Latin
| format = PDF
| accessdate = 2008-05-18
}}
* {{cite article
|first=V.
|last=Ramaswami
|title=Notes on Riemann's ζ-function
|year=1934
|journal=J. London Math. Soc.
|volume=9
|pages=165–169
|doi=10.1112/jlms/s1-9.3.165|issue=3}}
* {{citation
|first=Roger
|last=Apéry
|title=Irrationalité de ζ(2) et ζ(3)
|year=1979
|journal=Astérisque|volume=61|pages=11–13}}
* {{Citation
|author=A. van der Poorten
|title=A proof that Euler missed..
|journal=[[The Mathematical Intelligencer]]
|volume=1
|year=1979
|pages=195–203
|doi=10.1007/BF03028234
|url=http://www.maths.mq.edu.au/~alf/45.pdf
|issue=4}}
* {{cite journal
|journal=El. J. Combinat
|year=1996
|volume=3
|first1=Tewodoros
|last1=Amdeberhan
|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r13
|pages=#R13
|title=Faster and faster convergent series for ζ(3)
}}
*{{cite arxiv| first=D.J.| last=Broadhurst| eprint=math.CA/9803067| title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)| year=1998}}
* {{citation|first=Simon|last=Plouffe|url=http://www.lacim.uqam.ca/~plouffe/identities.html|title=Identities inspired from Ramanujan Notebooks II|year=1998}}
* {{citation|first=Simon|last=Plouffe|url=http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap97.html|title=Zeta(3) or Apery constant to 2000 places|year=undated}}
* {{citation|first=S.|last=Wedeniwski|title=The Value of Zeta(3) to 1,000,000 places|editor=Simon Plouffe|year=2001|publisher=Project Gutenberg}}
* {{citation
| last = Srivastava | first = H. M.
| year = 2000 | month = December
| title = Some Families of Rapidly Convergent Series Representations for the Zeta Functions
| url = http://www.math.nthu.edu.tw/~tjm/abstract/0012/tjm0012_3.pdf
| format = PDF
| journal = Taiwanese Journal of Mathematics
| volume = 4 | issue = 4 | pages = 569–598 
| oclc =36978119
| accessdate = 2008-05-18
}}
* {{cite web
|first1=Xavier
|last1=Gourdon
|first2=Pascal
|last2=Sebah
|url=http://numbers.computation.free.fr/Constants/Zeta3/zeta3.html
|title=The Apéry's constant: z(3)
|year=2003}}
* {{mathworld|title=Apéry's constant|urlname=AperysConstant}}
* {{citation|first1=Alexander J.|last1=Yee|first2=Raymond|last2=Chan|url=http://www.numberworld.org/nagisa_runs/computations.html|title=Large Computations|year=2009}}
 
{{PlanetMath attribution|id=4021|title=Apéry's constant}}
 
{{DEFAULTSORT:Aperys constant}}
[[Category:Mathematical constants]]
[[Category:Analytic number theory]]
[[Category:Irrational numbers]]
[[Category:Zeta and L-functions]]
 
[[ar:ثابتة أبيري]]
[[ca:Constant d'Apéry]]
[[da:Apérys konstant]]
[[de:Apéry-Konstante]]
[[es:Constante de Apéry]]
[[fr:Constante d'Apéry]]
[[ko:아페리 상수]]
[[it:Costante di Apéry]]
[[lmo:Custanta da Apéry]]
[[ja:アペリーの定数]]
[[pl:Stała Apéry'ego]]
[[pt:Constante de Apéry]]
[[ru:Постоянная Апери]]
[[tr:Apéry sabiti]]
[[zh:阿培里常数]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

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