Apéry's Constant

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

Apéry's constant is defined by

 (1)

(Sloane's A002117) where is the Riemann Zeta Function. Apéry (1979) proved that is Irrational, although it is not known if it is Transcendental. The Continued Fraction for is [1, 4, 1, 18, 1, 1, 1, 4, 1, ...] (Sloane's A013631). The positions at which the numbers 1, 2, ... occur in the continued fraction are 1, 12, 25, 2, 64, 27, 17, 140, 10, ... (Sloane's A033165).

Sums related to are

 (2)

(used by Apéry), and
 (3)

 (4)

 (5)

 (6)

where is the Dirichlet Lambda Function. The above equations are special cases of a general result due to Ramanujan (Berndt 1985). Apéry's proof relied on showing that the sum
 (7)

where is a Binomial Coefficient, satisfies the Recurrence Relation

 (8)

(van der Poorten 1979, Zeilberger 1991).

Apéry's constant is also given by

 (9)

where is a Stirling Number of the First Kind. This can be rewritten as
 (10)

where is the th Harmonic Number. Yet another expression for is
 (11)

(Castellanos 1988).

Integrals for include

 (12) (13)

Gosper (1990) gave
 (14)

A Continued Fraction involving Apéry's constant is
 (15)

(Apéry 1979, Le Lionnais 1983). Amdeberhan (1996) used Wilf-Zeilberger Pairs with
 (16)

to obtain
 (17)

For ,
 (18)

and for ,

 (19)

(Amdeberhan 1996). The corresponding for and 2 are
 (20)

and

 (21)

Gosper (1996) expressed as the Matrix Product
 (22)

where

 (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)

, and , giving the rapidly converging
 (30)

(Amdeberhan and Zeilberger 1997). The record as of Aug. 1998 was 64 million digits (Plouffe).

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.