Apéry's constant

In mathematics, Apéry's constant is a curious number that occurs in a variety of situations. It rises 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 $\zeta (3)$,

$\zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{i^{3}}}=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+\cdots$

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

$\zeta (3)=1.20205\;69031\;59594\;28539\;97381\;61511\;44999\;07649\;86292\,\ldots$ (sequence A002117 in the OEIS)

The reciprocal of this number 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 < N chosen uniformly at random will be relatively prime approaches this value).

List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
Binary	1.001100111011101...
Decimal	1.2020569031595942854...
Hexadecimal	1.33BA004F00621383...
Continued fraction	$1+{\frac {1}{4+{\frac {1}{1+{\frac {1}{18+{\frac {1}{\ddots \qquad {}}}}}}}}}$ Note that this continuing fraction is not periodic.

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

Series representation

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

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

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):

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

and

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

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

Many additional series representations have been found, including:

$\zeta (3)={\frac {8}{7}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)^{3}}}$

$\zeta (3)={\frac {4}{3}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(k+1)^{3}}}$

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

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

$\zeta (3)={\frac {8}{7}}-{\frac {8}{7}}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{t}\,2^{-5+12\,t}\,t\,\left(-3+9\,t+148\,t^{2}-432\,t^{3}-2688\,t^{4}+7168\,t^{5}\right)\,{t!}^{3}\,{\left(-1+2\,t\right)!}^{6}}{{\left(-1+2\,t\right)}^{3}\,\left(3\,t\right)!\,{\left(1+4\,t\right)!}^{3}}}$

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

and

$\zeta (3)=\sum _{k=0}^{\infty }(-1)^{n}{\frac {P(n)}{24}}{\frac {((2n+1)!(2n)!n!)^{3}}{(3n+2)!((4n+3)!)^{3}}}$

where

$P(n)=126392n^{5}+412708n^{4}+531578n^{3}+336367n^{2}+104000n+12463.\,$

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

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

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
January 2007	2,000,000,000	Howard Cheng, Guillaume Hanrot, Emmanuel Thomé, Eugene Zima & Paul Zimmermann
April 2006	10,000,000,000	Shigeru Kondo & Steve Pagliarulo (see Gourdon & Sebah (2003))
February 2003	1,000,000,000	Patrick Demichel & Xavier Gourdon
February 2002	600,001,000	Shigeru Kondo & Xavier Gourdon
September 2001	200,001,000	Shigeru Kondo & Xavier Gourdon
December 1998	128,000,026	Sebastian Wedeniwski (Wedeniwski 2001)
February 1998	14,000,074	Sebastian Wedeniwski
May 1997	10,536,006	Patrick Demichel
1997	1,000,000	Bruno Haible & Thomas Papanikolaou
1996	520,000	Greg J. Fee & Simon Plouffe
1887	32	Thomas Joannes Stieltjes
unknown	16	Adrien-Marie Legendre

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: 165–169.
• Apéry, Roger (1979), "Irrationalité de ζ(2) et ζ(3)", Astérisque, 61: 11–13.
• van der Poorten, Alfred (1979), "A proof that Euler missed. Apéry's proof of the irrationality of ζ(3). An informal report", Math. Intell., 1: 195–203.
• Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II
• Plouffe, Simon (undated), Zeta(3) or Apery constant to 2000 places {{citation}}: Check date values in: |year= (help).
• Wedeniwski, S. (2001), Simon Plouffe (ed.), The Value of Zeta(3) to 1,000,000 places, Project Gutenberg
• Srivastava, H. M. (2000), "Some Families of Rapidly Convergent Series Representations for the Zeta Functions" (PDF), Taiwanese Journal of Mathematics, 4 (4), Mathematical Society of the Republic of China (Taiwan): 569–598, ISSN 1027-5487, OCLC 36978119, retrieved 2008-05-18 {{citation}}: Unknown parameter |month= ignored (help)
• Euler, Leonhard (1773), "Exercitationes analyticae" (PDF), Novi Commentarii academiae scientiarum Petropolitanae (in Latin), 17: 173–204, retrieved 2008-05-18
• Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: z(3)
• Weisstein, Eric W. "Apéry's constant". MathWorld.

Apéry's constant at PlanetMath.