Apéry's constant
From Freepedia
In mathematics, Apéry's constant is a curious number that occurs in a variety of situations. It is defined as the number ζ(3),
- <math>\zeta(3)=1+\frac{1}{2^3} + \frac{1}{3^3} +\frac{1}{4^3} + \ldots</math>
where ζ is the Riemann zeta function. This value was named Apéry's constant when Roger Apéry proved Apéry's theorem in 1977, showing it to be irrational. It has a value of
- <math>\zeta(3)=1.20205\; 69031\; 59594\; 28539\; 97381\;
61511\; 44999\; 07649\; 86292\,\ldots</math>
Series representation
In 1772, Leonhard Euler give the series representation
- <math>\zeta(3)=\frac{\pi^2}{7}
\left[ 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {(2k+1)(2k+2) 2^{2k}} \right]</math>
which was subsequently rediscovered several times, including by Ramaswami in 1934.
Simon Plouffe gives several series, which are notable in that they can provide several digits of accuracy per iteration, and are thus very useful for high-precision calculations of Apéry's constant. These include:
- <math>\zeta(3)=\frac{7}{180}\pi^3 -2
\sum_{n=1}^\infty \frac{1}{n^3 (e^{2\pi n} -1)}</math>
and
- <math>\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)} </math>
Similar relations for the values of <math>\zeta(2n+1)</math> are given in the article zeta constants.
References
- 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).
This article incorporates material from Apéry's constant on PlanetMath, which is licensed under the GFDL.
