Note that this continued fraction is infinite, but it is not known whether this continued fraction is periodic or not.
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. It is defined as the number ζ(3),
The reciprocal of this constant 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).
This value was named for Roger Apéry (1916–1994), who in 1978 proved it to be 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.
Work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2n+1) must be irrational, and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.
which was subsequently rediscovered several times.
Similar relations for the values of are given in the article zeta constants.
Many additional series representations have been found, including:
Some of these have been used to calculate Apéry's constant with several million digits.
There are also numerous integral representations for the Apéry's constant. These include a simple formula
which follows from summation representation for the Apéry's constant. Other simple formulas include
which follow directly from the well-known integral formulas for the Riemann zeta function, as well as more complicated representations such as
see F. Beukers, or
see Iaroslav Blagouchine. Moreover, the 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 increase of 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 (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 Gourdon & Sebah (2003))|
|January 2009||15,510,000,000||Alexander J. Yee & Raymond Chan (see Yee & Chan (2009))|
|March 2009||31,026,000,000||Alexander J. Yee & Raymond Chan (see Yee & Chan (2009))|
|September 2010||100,000,001,000||Alexander J. Yee (see Yee)|
|September 2013||200,000,001,000||Robert J. Setti (Apéry's Constant - Zeta(3) - 200 Billion Digits)|
- T. Rivoal (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, Bibcode:2000CRASM.331..267R, doi:10.1016/S0764-4442(00)01624-4.
- W. Zudilin (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.
- Bruce C. Berndt, Ramanujan's notebooks, Part II (1989), Springer-Verlag. See chapter 14, formulas 25.1 and 25.3
- Johan Ludwig William Valdemar Jensen. Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver. L'Intermédiaire des mathématiciens, tome II, pp. 346-347, 1895.
- F. Beukers A Note on the Irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11, pp. 268-272, 1979.
- Iaroslav V. Blagouchine Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. PDF
- M. A. Evgrafov and K. A. Bezhanov and Y. V. Sidorov and M. V. Fedoriuk and M. I. Shabunin. A Collection of Problems in the Theory of Analytic Functions [in Russian]. Nauka, Moscow, 1969.
- Euler, Leonhard (1773), "Exercitationes analyticae" (PDF), Novi Commentarii academiae scientiarum Petropolitanae (in Latin) 17: 173–204, retrieved 2008-05-18
- 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
- Apéry, Roger (1979), "Irrationalité de ζ(2) et ζ(3)", Astérisque 61: 11–13
- A. van der Poorten (1979), "A proof that Euler missed.." (PDF), The Mathematical Intelligencer 1 (4): 195–203, doi:10.1007/BF03028234
- Amdeberhan, Tewodoros (1996), "Faster and faster convergent series for ζ(3)", El. J. Combinat 3: #R13
- Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), arXiv:math.CA/9803067
- Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II
- Plouffe, Simon, Zeta(3) or Apery constant to 2000 places
- Wedeniwski, S. (2001), Simon Plouffe, ed., The Value of Zeta(3) to 1,000,000 places, Project Gutenberg
- Srivastava, H. M. (December 2000), "Some Families of Rapidly Convergent Series Representations for the Zeta Functions" (PDF), Taiwanese Journal of Mathematics 4 (4): 569–598, OCLC 36978119, retrieved 2008-05-18
- Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: z(3)
- Weisstein, Eric W., "Apéry's constant", MathWorld.
- Yee, Alexander J.; Chan, Raymond (2009), Large Computations