Apéry's constant

List of numbers – Irrational and suspected irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δ_S – α – $e$ – $π$ – δ
Binary	1.001100111011101...
Decimal	1.2020569031595942854...
Hexadecimal	1.33BA004F00621383...
Continued fraction	$1 %2B \frac{1}{4 %2B \cfrac{1}{1 %2B \cfrac{1}{18 %2B \cfrac{1}{\ddots\qquad{}}}}}$ Note that this continuing fraction is not periodic.

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),

$\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}=1%2B\frac{1}{2^3} %2B \frac{1}{3^3} %2B \frac{1}{4^3} %2B \frac{1}{5^3} %2B \frac{1}{6^3} %2B \frac{1}{7^3} %2B \frac{1}{8^3} %2B \frac{1}{9^3} %2B \cdots\,\!$

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).

1 Apéry's theorem
2 Series representation
3 Other formulas
4 Known digits
5 Notes
6 References

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)):

$\zeta(3)=\frac{\pi^2}{7} \left[ 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {(2k%2B1)(2k%2B2) 2^{2k}} \right]$

which was subsequently rediscovered several times.

Ramanujan gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include^[3]:

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

Simon Plouffe has developed other series(Plouffe 1998):

$\zeta(3)= 14 \sum_{k=1}^\infty \frac{1}{k^3 \sinh(\pi k)} -\frac{11}{2} \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)} -\frac{7}{2} \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} %2B1)}.$

Similar relations for the values of $\zeta(2n%2B1)$ are given in the article zeta constants.

Many additional series representations have been found, including:

$\zeta(3) = \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k%2B1)^3}$

$\zeta(3) = \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k%2B1)^3}$

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

$\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1} \frac{56k^2-32k%2B5}{(2k-1)^2} \frac{((k-1)!)^3}{(3k)!}$

$\zeta(3)=\frac{8}{7}-\frac{8}{7}\sum_{k=1}^\infty \frac{{\left( -1 \right) }^k\,2^{-5 %2B 12\,k}\,k\, \left( -3 %2B 9\,k %2B 148\,k^2 - 432\,k^3 - 2688\,k^4 %2B 7168\,k^5 \right) \, {k!}^3\,{\left( -1 %2B 2\,k \right)�!}^6}{{\left( -1 %2B 2\,k \right) }^3\, \left( 3\,k \right)�!\,{\left( 1 %2B 4\,k \right)�!}^3}$

$\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{205k^2 %2B 250k %2B 77}{64} \frac{(k!)^{10}}{((2k%2B1)!)^5}$

and

$\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{P(k)}{24} \frac{((2k%2B1)!(2k)!k!)^3}{(3k%2B2)!((4k%2B3)!)^3}$

where

$P(k) = 126392k^5 %2B 412708k^4 %2B 531578k^3 %2B 336367k^2 %2B 104000k %2B 12463.\,$

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

$\zeta(3) = -\frac{1}{2} \, \psi^{(2)}(1).$

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
1735	16	Leonhard Euler
unknown	16	Adrien-Marie Legendre
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)

Notes

^ 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.
^ W. Zudilin (2001), "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational", Russ. Math. Surv. 56 (4): 774–776, doi:10.1070/RM2001v056n04ABEH000427.
^ Bruce C. Berndt, Ramanujan's notebooks, Part II (1989), Springer-Verlag. See chapter 14, formulas 25.1 and 25.3

References

Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), arXiv:math.CA/9803067
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..", 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 4 (4): 569–598, 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

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