Apéry's constant

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In mathematics, Apéry's constant is a curious number that occurs in a variety of situations. It is defined as the number ζ(3),

$\zeta(3)=1+\frac{1}{2^3} + \frac{1}{3^3} +\frac{1}{4^3} + \ldots$

where ζ is the Riemann zeta function. It has an approximate value of

$\zeta(3)=1.20205\; 69031\; 59594\; 28539\; 97381\; 61511\; 44999\; 07649\; 86292\,\ldots$

1 Apéry's theorem
2 Series representation
3 Other formulas
4 References

[edit] Apéry's theorem

This value was named for Roger Apéry (1916 - 1994), who in 1977 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.

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

[edit] Series representation

In 1772, Leonhard Euler gave the series representation

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

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:

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

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 $ζ(2 n + 1)$ 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 gives a series representation^[1] 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] References

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

V. Ramaswami, Notes on Riemann's ζ-function, (1934) J. London Math. Soc. 9 pp. 165-169.
Roger Apéry, Irrationalité de ζ(2) et ζ(3), (1979) Astérisque, 61:11-13.
Alfred van der Poorten, A proof that Euler missed. Apéry's proof of the irrationality of ζ(3). An informal report.,(1979) Math. Intell., 1:195-203.
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, (1998)
Simon Plouffe, Zeta(3) or Apery constant to 2000 places, (undated).
Xavier Gourdon & Pascal Sebah, The Apéry's constant: z(3)