*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Apéry's constant is defined by

(1) |

Sums related to are

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

Apéry's constant is also given by

(9) |

(10) |

(11) |

Integrals for include

(12) | |||

(13) |

Gosper (1990) gave

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

which gives 12 bits per term. The first few terms are

(24) | |||

(25) | |||

(26) |

which gives

(27) |

Given three Integers chosen at random, the probability that no common factor will divide them all is

(28) |

B. Haible and T. Papanikolaou computed to 1,000,000 Digits using a Wilf-Zeilberger
Pair identity with

(29) |

(30) |

**References**

Amdeberhan, T. ``Faster and Faster Convergent Series for .'' *Electronic J. Combinatorics* **3**, R13 1-2, 1996.
http://www.combinatorics.org/Volume_3/volume3.html#R13.

Amdeberhan, T. and Zeilberger, D. ``Hypergeometric Series Acceleration via the WZ Method.'' *Electronic J. Combinatorics* **4**, No. 2, R3, 1-3, 1997.
http://www.combinatorics.org/Volume_4/wilftoc.html#R03. Also available at
http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/accel.html.

Apéry, R. ``Irrationalité de et .'' *Astérisque* **61**, 11-13, 1979.

Berndt, B. C. *Ramanujan's Notebooks: Part I.* New York: Springer-Verlag, 1985.

Beukers, F. ``A Note on the Irrationality of .'' *Bull. London Math. Soc.* **11**, 268-272, 1979.

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, 1987.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' *Math. Mag.* **61**, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``The Great Enigma.'' In *The Book of Numbers.* New York: Springer-Verlag,
pp. 261-262, 1996.

Ewell, J. A. ``A New Series Representation for .'' *Amer. Math. Monthly* **97**, 219-220, 1990.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/apery/apery.html

Gosper, R. W. ``Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics.'' In *Computers in Mathematics*
(Ed. D. V. Chudnovsky and R. D. Jenks). New York: Marcel Dekker, 1990.

Haible, B. and Papanikolaou, T. ``Fast Multiprecision Evaluation of Series of Rational Numbers.'' Technical Report TI-97-7. Darmstadt, Germany: Darmstadt University of Technology, Apr. 1997.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 36, 1983.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Sloane, N. J. A. A013631, A033165, and A002117/M0020 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

van der Poorten, A. ``A Proof that Euler Missed... Apéry's Proof of the Irrationality of .'' *Math. Intel.* **1**,
196-203, 1979.

Zeilberger, D. ``The Method of Creative Telescoping.'' *J. Symb. Comput.* **11**, 195-204, 1991.

© 1996-9

1999-05-25