Apéry's constant

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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 ζ(3),

\zeta(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 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.

Contents

[edit] 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.

The result has remained quite isolated: little is known about ζ(n) for other odd numbers n.

[edit] 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_{n=1}^\infty \frac{1}{n^3 (e^{2\pi n} -1)}

and

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

Similar relations for the values of ζ(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_{n=1}^\infty (-1)^{n-1} \frac{(n!)^2}{n^3 (2n)!}
\zeta(3) = \frac{1}{4} \sum_{n=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_{t=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_{n=0}^\infty (-1)^n \frac{205n^2 + 250n + 77}{64} \frac{(n!)^{10}}{((2n+1)!)^5}

and

\zeta(3) = \sum_{n=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.

[edit] 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).

[edit] 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[1] Shigeru Kondo & Steve Pagliarulo
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

[edit] Notes

  1. ^ Claim made on Shigeru Kondo's website

[edit] References

This article incorporates material from Apéry's constant on PlanetMath, which is licensed under the GFDL.