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. It is defined as the number ζ(3),

where ζ is the Riemann zeta function. It has an approximate value of (Wedeniwski 2001)

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

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).

Apéry's theorem
Main article: Apéry's theorem

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,[1] and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[2]

Series representation

In 1772, Leonhard Euler (Euler 1773) gave the series representation (Srivastava 2000, p. 571 (1.11)):

which was subsequently rediscovered several times.

Simon Plouffe gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include (Plouffe 1998):

and

Similar relations for the values of ζ(2n + 1) are given in the article zeta constants.

Many additional series representations have been found, including:

and

where

Some of these have been used to calculate Apéry's constant with several million digits.

Broadhurst (1998) gives 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.

Other formulas

Apéry's constant can be expressed in terms of the second-order polygamma function as

It can be expressed as the non-periodic continued fraction [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (sequence A013631 in OEIS).

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.

Number of known decimal digits of Apéry's constant ζ(3)
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))

Apéry's constant

Notes

1. ^ T. Rivoal (2000), "La fonction zeta de Riemann prend une infnité de valuers irrationnelles aux entiers impairs", Comptes Rendus Acad. Sci. Paris Sér. I Math. 331: 267–270.
2. ^ W. Zudilin (2001), "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational", Russ. Math. Surv. 56: 774–776, doi:10.1070/RM2001v056n04ABEH000427.

References

* Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), arXiv (math.CA/9803067), http://arxiv.org/abs/math.CA/9803067
* Ramaswami, V. (1934), "Notes on Riemann's ζ-function", J. London Math. Soc. 9: 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...", The Mathematical Intelligencer 1 (4): 195–203, doi:10.1007/BF03028234, http://www.maths.mq.edu.au/~alf/45.pdf.
* Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II, http://www.lacim.uqam.ca/~plouffe/identities.html
* Plouffe, Simon (undated), Zeta(3) or Apery constant to 2000 places, http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap97.html
* 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 (Mathematical Society of the Republic of China (Taiwan)) 4 (4): 569–598, ISSN 1027-5487, OCLC 36978119, http://www.math.nthu.edu.tw/~tjm/abstract/0012/tjm0012_3.pdf, retrieved 2008-05-18
* Euler, Leonhard (1773), "Exercitationes analyticae" (in Latin) (PDF), Novi Commentarii academiae scientiarum Petropolitanae 17: 173–204, http://math.dartmouth.edu/~euler/docs/originals/E432.pdf, retrieved 2008-05-18
* Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: z(3), http://numbers.computation.free.fr/Constants/Zeta3/zeta3.html
* Weisstein, Eric W., "Apéry's constant" from MathWorld.
* Yee, Alexander J.; Chan, Raymond (2009), Large Computations, http://www.numberworld.org/nagisa_runs/computations.html