Note that this continued fraction is infinite, but it is not known whether this continued fraction is periodic or not.
This constant 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.
ζ(3) was named Apéry's constant for the French mathematician Roger Apéry, who proved in 1978 that it is irrational. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.
It is still not known whether Apéry's constant is transcendental.
which was subsequently rediscovered several times.
Other classical series representations include:
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, found by Amdeberhan in 1996, gives (asymptotically) 1.43 new correct decimal places per term:
The following series representation, found by Amdeberhan and Zeilberger in 1997, gives (asymptotically) 3.01 new correct decimal places per term:
The following series representation, found by Sebastian Wedeniwski in 1998, gives (asymptotically) 5.04 new correct decimal places per term:
It was used used by Wedeniwski to calculate Apéry's constant with several million correct decimal places.
The following series representation, found by Mohamud Mohammed in 2005, gives (asymptotically) 3.92 new correct decimal places per term:
The following series representation has an undetermined source:
Digit by digit
Srivastava collected many series that converge to Apéry's constant.
There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.
For example, this one follows from the summation representation for Apéry's constant:
The next two follow directly from the well-known integral formulas for the Riemann zeta function:
More complicated formulas
another by F. Beukers:
and yet another by Iaroslav Blagouchine:
Evgrafov et al.'s connection to the derivatives of the Г-function
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.
|Date||Decimal digits||Computation performed by|
|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|
|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|
|January 2009||15,510,000,000||Alexander J. Yee & Raymond Chan|
|March 2009||31,026,000,000||Alexander J. Yee & Raymond Chan|
|September 2010||100,000,001,000||Alexander J. Yee|
|September 2013||200,000,001,000||Robert J. Setti|
|August 2015||250,000,000,000||Ron Watkins|
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).
Extension to ζ(2n+1)
Many people have tried to extend Apéry's proof that ζ(3) is irrational to other odd zeta values. In 2000, Tanguy Rivoal showed that infinitely many of the numbers ζ(2n+1) must be irrational. In 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.
- See Wedeniwski 2001.
- See Apéry 1979.
- See van der Poorten 1979.
- See Beukers 1979.
- See Zudilin 2002.
- See Euler 1773.
- See Srivastava 2000, p. 571 (1.11).
- See Hjortnaes 1953.
- See Amdeberhan 1996.
- See Amdeberhan & Zeilberger 1997.
- See Wedeniwski 1998 and 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).
- See Wedeniwski 1998 and Wedeniwski 2001.
- See Mohammed 2005.
- See Broadhurst 1998.
- See Berndt 1989, chapter 14, formulas 25.1 and 25.3.
- See Plouffe 1998.
- See Srivastava 2000.
- See Jensen 1895.
- See Blagouchine 2014.
- See Evgrafov et al. 1969, exercise 30.10.1.
- See Gourdon & Sebah 2003.
- See Yee 2009.
- See Yee 2015.
- See Rivoal 2000.
- See Zudilin 2001.
- Amdeberhan, Tewodros (1996), "Faster and faster convergent series for ζ(3)", El. J. Combinat. 3 (1).
- Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", El. J. Combinat. 4 (2).
- Apéry, Roger (1979), "Irrationalité de ζ(2) et ζ(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) and ζ(3)", Bull. London Math. Soc. 11: 268–272.
- 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.
- Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(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.
- Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: ζ(3).
- Hjortnaes, M. M. (August 1953), Overføring av rekken 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 (Gauthier-Villars) II: 346–347.
- Mohammed, Mohamud (2005), "Infinite families of accelerated series for some classical constants by the Markov-WZ method", Discrete Mathematics and Theoretical Computer Science (Springer) 7: 11–24.
- Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II.
- Plouffe, Simon, Zeta(3) or Apery constant to 2000 places.
- Ramaswami, V. (1934), "Notes on Riemann's ζ-function", J. London Math. Soc. 9 (3): 165–169, doi:10.1112/jlms/s1-9.3.165.
- 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. Mathématique 331 (4): 267–270, arXiv:math/0008051, Bibcode:2000CRASM.331..267R, doi:10.1016/S0764-4442(00)01624-4.
- Setti, Robert J. (2015), Apéry's Constant - Zeta(3) - 200 Billion Digits.
- 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, 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)" (PDF), The Mathematical Intelligencer 1 (4): 195–203, doi:10.1007/BF03028234.
- 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).
- Yee, Alexander J. (2009), Large Computations.
- Yee, Alexander J. (2015), Zeta(3) - Apery's Constant
- Zudilin, Wadim (2001), "One of the numbers ζ(5), ζ(7), ζ(9), ζ(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 Apery's theorem, arXiv:math/0202159, Bibcode:2002math......2159Z.