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In mathematics, at the intersection of number theory and special functions, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

{\displaystyle {\begin{aligned}\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{aligned}}}
where ζ is the Riemann zeta function. It has an approximate value of[1]

ζ(3) = 1.202056903159594285399738161511449990764986292… (sequence A002117 in the OEIS).

The 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 trees[2] 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

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.[3] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[4] and simpler proofs were found later.[5]

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for $$\zeta (3)$$ ,

$${\displaystyle \zeta (3)=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz,}$$

by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that

$${\displaystyle I_{3}:=-{\frac {1}{2}}\int _{0}^{1}\int _{0}^{1}{\frac {P_{n}(x)P_{n}(y)\log(xy)}{1-xy}}\,dx\,dy=b_{n}\zeta (3)-a_{n},}$$

where $${\displaystyle |I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}}$$, $$P_{n}(z)$$ are the Legendre polynomials, and the subsequences $${\displaystyle b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z} }$$ are integers or almost integers.

It is still not known whether Apéry's constant is transcendental.
Series representations
Classical

In addition to the fundamental series:

$${\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},}$$

Leonhard Euler gave the series representation:[6]

$${\displaystyle \zeta (3)={\frac {\pi ^{2}}{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)}$$

in 1772, which was subsequently rediscovered several times.[7]

Other classical series representations include:

{\displaystyle {\begin{aligned}\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{aligned}}}

Fast convergence

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

The following series representation was found by A. A. Markov in 1890,[8] rediscovered by Hjortnaes in 1953,[9] and rediscovered once more and widely advertised by Apéry in 1979:[3]

$${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}}$$

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[10]

$${\displaystyle \zeta (3)={\frac {1}{4}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {(k-1)!^{3}(56k^{2}-32k+5)}{(2k-1)^{2}(3k)!}}}$$

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[11]

$${\displaystyle \zeta (3)={\frac {1}{64}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {k!^{10}(205k^{2}+250k+77)}{(2k+1)!^{5}}}}$$

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[12]

$${\displaystyle \zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}}$$

It has been used to calculate Apéry's constant with several million correct decimal places.[13]

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[14]

$${\displaystyle \zeta (3)={\frac {1}{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}}$$

Digit by digit

In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.[15]
Others

The following series representation was found by Ramanujan:[16]

$${\displaystyle \zeta (3)={\frac {7}{180}}\pi ^{3}-2\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}}$$

The following series representation was found by Simon Plouffe in 1998:[17]

$${\displaystyle \zeta (3)=14\sum _{k=1}^{\infty }{\frac {1}{k^{3}\sinh(\pi k)}}-{\frac {11}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}-{\frac {7}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}+1)}}.}$$

Srivastava (2000) 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:

$${\displaystyle \zeta (3)=\int _{0}^{1}\!\!\int _{0}^{1}\!\!\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz}.$$

The next two follow directly from the well-known integral formulas for the Riemann zeta function:

$${\displaystyle \zeta (3)={\frac {1}{2}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx}$$

and

$${\displaystyle \zeta (3)={\frac {2}{3}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}+1}}\,dx}.$$

This one follows from a Taylor expansion of χ3(eix) about x = ±π/2, where χν(z) is the Legendre chi function:

$${\displaystyle \zeta (3)={\frac {4}{7}}\int _{0}^{\frac {\pi }{2}}x\log {(\sec {x}+\tan {x})}\,dx}$$

Note the similarity to

$${\displaystyle G={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\log {(\sec {x}+\tan {x})}\,dx}$$

where G is Catalan's constant.
More complicated formulas

Other formulas include:[18]

$${\displaystyle \zeta (3)=\pi \!\!\int _{0}^{\infty }\!{\frac {\cos(2\arctan {x})}{\left(x^{2}+1\right)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx},$$

and,[19]

$${\displaystyle \zeta (3)=-{\frac {1}{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(xy)}{\,1-xy\,}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(1-xy)}{\,xy\,}}\,dx\,dy},$$

Mixing these two formulas, one can obtain :

$${\displaystyle \zeta (3)=\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,x\,}}\,dx},$$

By symmetry,

$${\displaystyle \zeta (3)=\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,1-x\,}}\,dx},$$

Summing both, $${\displaystyle \zeta (3)={\frac {1}{2}}\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,x(1-x)\,}}\,dx}.$$

Also,[20]

{\displaystyle {\begin{aligned}\zeta (3)&={\frac {8\pi ^{2}}{7}}\!\!\int _{0}^{1}\!{\frac {x\left(x^{4}-4x^{2}+1\right)\log \log {\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)\log \log {x}}{\,(1+x^{2})^{4}\,}}\,dx\end{aligned}}}.

A connection to the derivatives of the gamma function

$${\displaystyle \zeta (3)=-{\tfrac {1}{2}}\Gamma '''(1)+{\tfrac {3}{2}}\Gamma '(1)\Gamma ''(1)-{\big (}\Gamma '(1){\big )}^{3}=-{\tfrac {1}{2}}\,\psi ^{(2)}(1)}$$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma-functions.[21]
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 increasing performance of computers and to algorithmic improvements.

Number of known decimal digits of Apéry's constant ζ(3)
Date Decimal digits Computation performed by
1735 16 Leonhard Euler
1887 32 Thomas Joannes Stieltjes
1996 520000 Greg J. Fee & Simon Plouffe
1997 1000000 Bruno Haible & Thomas Papanikolaou
May 1997 10536006 Patrick Demichel
February 1998 14000074 Sebastian Wedeniwski
March 1998 32000213 Sebastian Wedeniwski
July 1998 64000091 Sebastian Wedeniwski
December 1998 128000026 Sebastian Wedeniwski[1]
September 2001 200001000 Shigeru Kondo & Xavier Gourdon
February 2002 600001000 Shigeru Kondo & Xavier Gourdon
February 2003 1000000000 Patrick Demichel & Xavier Gourdon[22]
April 2006 10000000000 Shigeru Kondo & Steve Pagliarulo
January 21, 2009 15510000000 Alexander J. Yee & Raymond Chan[23]
February 15, 2009 31026000000 Alexander J. Yee & Raymond Chan[23]
September 17, 2010 100000001000 Alexander J. Yee[24]
September 23, 2013 200000001000 Robert J. Setti[24]
August 7, 2015 250000000000 Ron Watkins[24]
December 21, 2015 400000000000 Dipanjan Nag[25]
August 13, 2017 500000000000 Ron Watkins[24]
May 26, 2019 1000000000000 Ian Cutress[26]
July 26, 2020 1200000000100 Seungmin Kim[27][28]

Reciprocal

The reciprocal of ζ(3) is the probability that any three positive integers, 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).[29]
Extension to ζ(2n + 1)
Main article: Particular values of Riemann zeta function

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the zeta function with odd arguments. Infinitely many of the numbers ζ(2n + 1) must be irrational,[30] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[31]

Riemann zeta function
Basel problem — ζ(2)
List of sums of reciprocals

Notes

Wedeniwski (2001).
Frieze (1985).
Apéry (1979).
van der Poorten (1979).
Beukers (1979); Zudilin (2002).
Euler (1773).
Srivastava (2000), p. 571 (1.11).
Markov (1890).
Hjortnaes (1953).
Amdeberhan (1996).
Amdeberhan & Zeilberger (1997).
Wedeniwski (1998); Wedeniwski (2001). In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from Amdeberhan & Zeilberger (1997). The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).
Wedeniwski (1998); Wedeniwski (2001).
Mohammed (2005).
Berndt (1989, chapter 14, formulas 25.1 and 25.3).
Plouffe (1998).
Jensen (1895).
Beukers (1979).
Blagouchine (2014).
Evgrafov et al. (1969), exercise 30.10.1.
Gourdon & Sebah (2003).
Yee (2009).
Yee (2017).
Nag (2015).
Records set by y-cruncher, retrieved June 8, 2019
Records set by y-cruncher, archived from the original on 2020-08-10, retrieved August 10, 2020
Apéry's constant world record by Seungmin Kim, retrieved July 28, 2020
Mollin (2009).
Rivoal (2000).

Zudilin (2001).

References

Amdeberhan, Tewodros (1996), "Faster and faster convergent series for ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3)", El. J. Combinat., 3 (1).
Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", El. J. Combinat., 4 (2), arXiv:math/9804121, Bibcode:1998math......4121A.
Apéry, Roger (1979), "Irrationalité de ζ 2 {\displaystyle \zeta 2} {\displaystyle \zeta 2} et ζ 3 {\displaystyle \zeta 3} {\displaystyle \zeta 3}", Astérisque, 61: 11–13.
Berndt, Bruce C. (1989), Ramanujan's notebooks, Part II, Springer.
Beukers, F. (1979), "A Note on the Irrationality of ζ ( 2 ) {\displaystyle \zeta (2)} \zeta (2) and ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3)", Bull. London Math. Soc., 11 (3): 268–272, doi:10.1112/blms/11.3.268.
Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results", The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5, S2CID 120943474.
Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3) and ζ ( 5 ) {\displaystyle \zeta (5)} \zeta (5), arXiv:math.CA/9803067.
Euler, Leonhard (1773), "Exercitationes analyticae" (PDF), Novi Commentarii Academiae Scientiarum Petropolitanae (in Latin), 17: 173–204, retrieved 2008-05-18.
Evgrafov, M. A.; Bezhanov, K. A.; Sidorov, Y. V.; Fedoriuk, M. V.; Shabunin, M. I. (1969), A Collection of Problems in the Theory of Analytic Functions [in Russian], Moscow: Nauka.
Frieze, A. M. (1985), "On the value of a random minimum spanning tree problem", Discrete Applied Mathematics, 10 (1): 47–56, doi:10.1016/0166-218X(85)90058-7, MR 0770868.
Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3).
Hjortnaes, M. M. (August 1953), Overføring av rekken ∑ k = 1 ∞ ( 1 k 3 ) {\displaystyle \sum _{k=1}^{\infty }\left({\frac {1}{k^{3}}}\right)} {\displaystyle \sum _{k=1}^{\infty }\left({\frac {1}{k^{3}}}\right)} til et bestemt integral, in Proc. 12th Scandinavian Mathematical Congress, Lund, Sweden: Scandinavian Mathematical Society, pp. 211–213.
Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347.
Markov, A. A. (1890), "Mémoire sur la transformation des séries peu convergentes en séries très convergentes", Mém. De l'Acad. Imp. Sci. De St. Pétersbourg, t. XXXVII, No. 9: 18pp.
Mohammed, Mohamud (2005), "Infinite families of accelerated series for some classical constants by the Markov-WZ method", Discrete Mathematics and Theoretical Computer Science, 7: 11–24.
Mollin, Richard A. (2009), Advanced Number Theory with Applications, Discrete Mathematics and Its Applications, CRC Press, p. 220, ISBN 9781420083293.
Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II.
Rivoal, Tanguy (2000), "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs", Comptes Rendus de l'Académie des Sciences, Série I, 331 (4): 267–270, arXiv:math/0008051, Bibcode:2000CRASM.331..267R, doi:10.1016/S0764-4442(00)01624-4, S2CID 119678120.
Srivastava, H. M. (December 2000), "Some Families of Rapidly Convergent Series Representations for the Zeta Functions" (PDF), Taiwanese Journal of Mathematics, 4 (4): 569–599, doi:10.11650/twjm/1500407293, OCLC 36978119, retrieved 2015-08-22.
van der Poorten, Alfred (1979), "A proof that Euler missed ... Apéry's proof of the irrationality of ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3)" (PDF), The Mathematical Intelligencer, 1 (4): 195–203, doi:10.1007/BF03028234, S2CID 121589323, archived from the original (PDF) on 2011-07-06.
Wedeniwski, Sebastian (2001), Simon Plouffe (ed.), The Value of Zeta(3) to 1,000,000 places, Project Gutenberg (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).
Wedeniwski, Sebastian (13 December 1998), The Value of Zeta(3) to 1,000,000 places (Message to Simon Plouffe, with original text but only some decimal places).
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Yee, Alexander J. (2017), Zeta(3) - Apéry's Constant

Nag, Dipanjan (2015), Calculated Apéry's constant to 400,000,000,000 Digit, A world record
Zudilin, Wadim (2001), "One of the numbers ζ ( 5 ) {\displaystyle \zeta (5)} \zeta (5), ζ ( 7 ) {\displaystyle \zeta (7)} {\displaystyle \zeta (7)}, ζ ( 9 ) {\displaystyle \zeta (9)} {\displaystyle \zeta (9)}, ζ ( 11 ) {\displaystyle \zeta (11)} {\displaystyle \zeta (11)} is irrational", Russ. Math. Surv., 56 (4): 774–776, Bibcode:2001RuMaS..56..774Z, doi:10.1070/RM2001v056n04ABEH000427.
Zudilin, Wadim (2002), An elementary proof of Apéry's theorem, arXiv:math/0202159, Bibcode:2002math......2159Z.

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Weisstein, Eric W., "Apéry's constant", Mathworld
Plouffe, Simon, Zeta(3) or Apéry constant to 2000 places
Setti, Robert J. (2015), Apéry's Constant - Zeta(3) - 200 Billion Digits, archived from the original on 2013-10-08.

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Irrational numbers

Chaitin's (Ω) Liouville Prime (ρ) Logarithm of 2 Gauss's (G) Twelfth root of 2 Apéry's (ζ(3)) Plastic (ρ) Square root of 2 Supergolden ratio (ψ)

Erdős–Borwein (E) Golden ratio (φ) Square root of 3 Square root of 5 Silver ratio (δS) Euler's (e) Pi (π)

Gold, square root of 2, and square root of 3 rectangles.png

Schizophrenic Transcendental Trigonometric