View source for Apéry's constant

{| class="infobox bordered" width = 55 cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| {{gaps|1.0011|0011|1011|1010|…}}
|-
| [[Decimal]]
| {{gaps|1.20205|69031|59594|2854…}}
|-
| [[Hexadecimal]]
| {{gaps|1.33BA|004F|0062|1383|…}}
|-
| [[Continued fraction]]
| <math>1 + \frac{1}{4 + \cfrac{1}{1 + \cfrac{1}{18 + \cfrac{1}{\ddots\qquad{}}}}}</math><br><small>Note that this continued fraction is infinite, but it is not known whether this continued fraction is [[Periodic continued fraction|periodic]] or not. </small>
|}

In [[mathematics]], at the intersection of [[number theory]] and [[special function]]s, '''Apéry's constant''' is defined as the number

:<math>\begin{align}\zeta(3) &= \sum_{n=1}^\infty\frac{1}{n^3} \\&= \lim_{n \to \infty}\left(\frac{1}{1^3} + \frac{1}{2^3} + \cdots + \frac{1}{n^3}\right)\end{align}</math>

where {{math|''ζ''}} is the [[Riemann zeta function]]. It has an approximate value of<ref name="Wedeniwski 2001">See {{harvnb|Wedeniwski|2001}}.</ref>

:{{math|''ζ''(3) {{=}} {{gaps|1.20205|69031|59594|28539|97381|61511|44999|07649|86292|…}}}} {{OEIS|id=A002117}}.

The [[constant (mathematics)|constant]] is named after [[Roger Apéry]]. 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 the analysis of [[random minimum spanning tree]]s<ref>See {{harvnb|Frieze|1985}}.</ref> and 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]].

==Irrational number==
{{math|''ζ''(3)}} was named ''Apéry's constant'' after the French mathematician [[Roger Apéry]], who proved in 1978 that it is an [[irrational number]].<ref name="Apery-1979">See {{harvnb|Apéry|1979}}.</ref> This result is known as ''[[Apéry's theorem]]''. The original proof is complex and hard to grasp,<ref>See {{harvnb|van der Poorten|1979}}.</ref> and simpler proofs were found later.<ref name="Beukers 1979">See {{harvnb|Beukers|1979}}.</ref><ref>See {{harvnb|Zudilin|2002}}.</ref>

Beuker's simplified irrationality proof involves approximating the integrand of the known triple integral for <math>\zeta(3)</math>,

:<math>\zeta(3) = \int_0^1 \int_0^1 \int_0^1 \frac{1}{1-xyz}\, dx\, dy\, dz,</math> 
by the [[Legendre polynomials]].
In particular, van der Poorten's article chronicles this approach by noting that

:<math>I_3 := -\frac{1}{2} \int_0^1 \int_0^1 \frac{P_n(x) P_n(y) \log(xy)}{1-xy}\, dx\, dy = b_n \zeta(3) - a_n, </math>

where <math>|I| \leq \zeta(3) (1-\sqrt{2})^{4n}</math>, <math>P_n(z)</math> are the [[Legendre polynomials]], and the [[subsequences]] <math>b_n, 2 \operatorname{lcm}(1,2,\ldots,n) \cdot a_n \in \mathbb{Z}</math> are integers or [[almost integer]]s.

It is still not known whether Apéry's constant is [[transcendental number|transcendental]].

==Series representations==

===Classical===
In addition to the fundamental series: 
:<math>\zeta(3)=\sum_{k=1}^\infty \frac{1}{k^3},</math>
[[Leonhard Euler]] gave the series representation:<ref>See {{harvnb|Euler|1773}}.</ref>
:<math>\zeta(3)=\frac{\pi^2}{7} \left( 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {2^{2k}(2k+1)(2k+2)} \right)</math>
in 1772, which was subsequently rediscovered several times.<ref>See {{harvnb|Srivastava|2000|loc=p. 571 (1.11)}}.</ref>

Other classical series representations include:
:<math>\begin{align} \zeta(3) &= \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k+1)^3} \\ \zeta(3) &= \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^3} \end{align}</math>

===Fast convergence===
Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of {{math|''ζ''(3)}}. Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "[[#Known digits|Known digits]]").

The following series representation was found by Hjortnaes in 1953,<ref>See {{harvnb|Hjortnaes|1953}}.</ref> then rediscovered and widely advertised by Apéry in 1979:<ref name="Apery-1979" />
:<math>\zeta(3) = -\frac{5}{2} \sum_{k=1}^\infty (-1)^k \frac{k!^2}{(2k)!k^3}</math>

The following series representation, found by Amdeberhan in 1996,<ref>See {{harvnb|Amdeberhan|1996}}.</ref> gives (asymptotically) 1.43 new correct decimal places per term:
:<math>\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1} \frac{(k-1)!^3(56k^2 - 32k + 5)}{(2k-1)^2(3k)!}</math>

The following series representation, found by Amdeberhan and Zeilberger in 1997,<ref>See {{harvnb|Amdeberhan|Zeilberger|1997}}.</ref> gives (asymptotically) 3.01 new correct decimal places per term:
:<math>\zeta(3) = \frac{1}{64} \sum_{k=0}^\infty (-1)^k \frac{k!^{10}(205k^2 + 250k + 77)}{(2k+1)!^5}</math>

The following series representation, found by Sebastian Wedeniwski in 1998,<ref>See {{harvnb|Wedeniwski|1998}} and {{harvnb|Wedeniwski|2001}}. In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from {{harvnb|Amdeberhan|Zeilberger|1997}}. The discovery year (1998) is mentioned in [http://plouffe.fr/simon/articles/TableofRecords.pdf Simon Plouffe's Table of Records] (8 April 2001).</ref> gives (asymptotically) 5.04 new correct decimal places per term:
:<math>\zeta(3) = \frac{1}{24} \sum_{k=0}^\infty (-1)^k \frac{(2k+1)!^3(2k)!^3k!^3(126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463)}{(3k+2)!(4k+3)!^3}</math>

It was used by Wedeniwski to calculate Apéry's constant with several million correct decimal places.<ref>See {{harvnb|Wedeniwski|1998}} and {{harvnb|Wedeniwski|2001}}.</ref>

The following series representation, found by Mohamud Mohammed in 2005,<ref>See {{harvnb|Mohammed|2005}}.</ref> gives (asymptotically) 3.92 new correct decimal places per term:
:<math>\zeta(3) = \frac{1}{2}\,\sum_{k=0}^\infty \frac{(-1)^k(2k)!^3(k+1)!^6(40885k^5 + 124346k^4 + 150160k^3 + 89888k^2 + 26629k + 3116)}{(k+1)^2(3k+3)!^4}</math>
where

===Digit by digit===
In 1998, Broadhurst<ref>See {{harvnb|Broadhurst|1998}}.</ref> gave 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]].

===Others===
The following series representation was found by [[Ramanujan]]:<ref>See {{harvnb|Berndt|1989|loc=chapter 14, formulas 25.1 and 25.3}}.</ref>
:<math>\zeta(3)=\frac{7}{180}\pi^3 -2 \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}</math>

The following series representation was found by [[Simon Plouffe]] in 1998:<ref>See {{harvnb|Plouffe|1998}}.</ref>
:<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>

Srivastava<ref>See {{harvnb|Srivastava|2000}}.</ref> collected many series that converge to Apéry's constant.

==Integral representations==
There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

===Simple formulas===
For example, this one follows from the summation representation for Apéry's constant:
:<math>\zeta(3) =\int_0^1\!\! \int_0^1\!\! \int_0^1 \frac{1}{1-xyz}\, dx\,dy\,dz </math>.

The next two follow directly from the well-known integral formulas for the [[Riemann zeta function]]:
:<math>\zeta(3) =\frac{1}{2}\int_0^\infty \frac{x^2}{e^x-1}\, dx </math>
and
:<math>\zeta(3) =\frac{2}{3}\int_0^\infty \frac{x^2}{e^x+1}\, dx </math>.

This one follows from a Taylor expansion of {{math|''χ''<sub>3</sub>(''e<sup>ix</sup>'')}} about {{math|''x'' {{=}} ±{{sfrac|π|2}}}}, where {{math|''χ''<sub>''ν''</sub>(''z'')}} is the [[Legendre chi function]]:
:<math>\zeta(3) =\frac{4}{7}\int_0^\frac{\pi}{2} x \log{(\sec{x}+\tan{x})}\, dx </math>
Note the similarity to
:<math>G =\frac{1}{2}\int_0^\frac{\pi}{2} \log{(\sec{x}+\tan{x})}\, dx </math>
where {{mvar|G}} is [[Catalan's constant]].

===More complicated formulas===
For example, one formula was found by [[Johan Jensen (mathematician)|Johan Jensen]]:<ref>See {{harvnb|Jensen|1895}}.</ref>
:<math>\zeta(3)=\pi\!\!\int_{0}^{\infty} \! \frac{\cos(2\arctan{x})}{\left(x^2+1\right)\left(\cosh\frac{1}{2}\pi x\right)^2}\, dx</math>,
another by F. Beukers:<ref name="Beukers 1979"/>
:<math>\zeta(3) =-\frac{1}{2}\int_0^1 \!\!\int_0^1 \frac{\ln(xy)}{\,1-xy\,}\, dx \, dy = -\int_0^1 \!\!\int_0^1 \frac{\ln(1-xy)}{\,xy\,}\, dx \, dy</math>,
Mixing these two formula, one can obtain :
<math>\zeta(3) =\int_0^1 \!\! \frac{\ln(x)\ln(1-x)}{\,x\,}\, dx </math>

and yet another by Iaroslav Blagouchine:<ref name="iaroslav_06">See {{harvnb|Blagouchine|2014}}.</ref>
:<math>\begin{align} \zeta(3) &= \frac{8\pi^2}{7}\!\!\int_0^1 \! \frac{x\left(x^4-4x^2+1\right)\ln\ln\frac{1}{x}}{\,(1+x^2)^4\,}\, dx \\&= \frac{8\pi^2}{7}\!\!\int_1^\infty \!\frac{x\left(x^4-4x^2+1\right)\ln\ln{x}}{\,(1+x^2)^4\,}\, dx\end{align}</math>.
Evgrafov et al.'s connection to the derivatives of the [[gamma function]] 
:<math>\zeta(3) = -\tfrac{1}{2}\Gamma'''(1)+\tfrac{3}{2}\Gamma'(1)\Gamma''(1)- \big(\Gamma'(1)\big)^3 = -\tfrac{1}{2} \, \psi^{(2)}(1)</math>
is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and [[Polygamma function|polygamma-functions]].<ref>See {{harvnb|Evgrafov|Bezhanov|Sidorov|Fedoriuk|1969|loc=exercise 30.10.1}}.</ref>

==Known digits==
The number of known digits of Apéry's constant {{math|''ζ''(3)}} has increased dramatically during the last decades. This is due both to the increasing performance of computers and to algorithmic improvements.

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of Apéry's constant {{math|''ζ''(3)}}
! Date || Decimal digits || Computation performed by
|-
| 1735 ||align="right"| 16 || [[Leonhard Euler]]
|-
| unknown ||align="right"| 16 || [[Adrien-Marie Legendre]]
|-
| 1887 ||align="right"| 32 || [[Thomas Joannes Stieltjes]]
|-
| 1996 ||align="right"| {{val|520000}} || Greg J. Fee & [[Simon Plouffe]]
|-
| 1997 ||align="right"| {{val|1000000}} || Bruno Haible & Thomas Papanikolaou
|-
| May 1997 ||align="right"| {{val|10536006}} || Patrick Demichel
|-
| February 1998 ||align="right"| {{val|14000074}} || Sebastian Wedeniwski
|-
| March 1998 ||align="right"| {{val|32000213}} || Sebastian Wedeniwski
|-
| July 1998 ||align="right"| {{val|64000091}} || Sebastian Wedeniwski
|-
| December 1998 ||align="right"| {{val|128000026}} || Sebastian Wedeniwski<ref name="Wedeniwski 2001"/>
|-
| September 2001 ||align="right"| {{val|200001000}} || Shigeru Kondo & Xavier Gourdon
|-
| February 2002 ||align="right"| {{val|600001000}} || Shigeru Kondo & Xavier Gourdon
|-
| February 2003 ||align="right"| {{val|1000000000}} || Patrick Demichel & Xavier Gourdon<ref>See {{harvnb|Gourdon|Sebah|2003}}.</ref>
|-
| April 2006 ||align="right"| {{val|10000000000}} || Shigeru Kondo & Steve Pagliarulo
|-
| January 2009 ||align="right"| {{val|15510000000}} || Alexander J. Yee & Raymond Chan<ref name="Yee-2009">See {{harvnb|Yee|2009}}.</ref>
|-
| March 2009 ||align="right"| {{val|31026000000}} || Alexander J. Yee & Raymond Chan<ref name="Yee-2009" />
|-
| September 2010 ||align="right"| {{val|100000001000}} || Alexander J. Yee<ref name="Yee-2017">See {{harvnb|Yee|2017}}.</ref>
|-
| September 2013 ||align="right"| {{val|200000001000}} || Robert J. Setti<ref name="Yee-2017" />
|-
| August 2015 ||align="right"| {{val|250000000000}} || Ron Watkins<ref name="Yee-2017" />
|-
| November 2015 ||align="right"| {{val|400000000000}} || Dipanjan Nag<ref name="Nag-2015">See {{harvnb|Nag|2015}}.</ref>
|-
| August 2017 ||align="right"| {{val|500000000000}} || Ron Watkins<ref name="Yee-2017" />
|-
| June 2019 ||align="right"| {{val|1000000000000}} || Ian Curtess<ref>{{Cite web|url=http://www.numberworld.org/y-cruncher/records.html|title=Records set by y-cruncher|access-date=June 8, 2019}}</ref>
|}

==Reciprocal==
The [[reciprocal (mathematics)|reciprocal]] of {{math|''ζ''(3)}} is the [[probability]] that any three [[positive integer]]s, chosen at random, will be [[relatively prime]] (in the sense that as {{math|''N''}} goes to infinity, the probability that three positive integers less than {{math|''N''}} chosen uniformly at random will be relatively prime approaches this value).{{sfnp|Mollin|2009}}

==Extension to {{math|''ζ''(2''n'' + 1)}}==
{{main|Particular values of Riemann zeta function}}
Many people have tried to extend Apéry's proof that {{math|''ζ''(3)}} is irrational to other values of the zeta function with odd arguments. In 2000, Tanguy Rivoal showed that infinitely many of the numbers {{math|''ζ''(2''n'' + 1)}} must be irrational.<ref>See {{harvnb|Rivoal|2000}}.</ref> In 2001, [[Wadim Zudilin]] proved that at least one of the numbers {{math|''ζ''(5)}}, {{math|''ζ''(7)}}, {{math|''ζ''(9)}}, and {{math|''ζ''(11)}} must be irrational.<ref>See {{harvnb|Zudilin|2001}}.</ref>

==See also==
*[[Riemann zeta function]]
*[[Basel problem]] — {{math|''ζ''(2)}}
*[[List of sums of reciprocals]]

==Notes==
{{Reflist|30em}}

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{{refend}}

==Credits==
{{PlanetMath attribution|urlname=AperysConstant|title=Apéry's constant}}

{{Irrational number}}

{{DEFAULTSORT:Aperys constant}}
[[Category:Mathematical constants]]
[[Category:Analytic number theory]]
[[Category:Irrational numbers]]
[[Category:Zeta and L-functions]]