| 释义 |
Apéry's ConstantN.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Apéry's constant is defined by
 | (1) |
 is the Riemann Zeta Function. Apéry (1979) proved that  isIrrational, although it is not known if it is Transcendental. TheContinued Fraction for is [1, 4, 1, 18, 1, 1, 1, 4, 1, ...] (Sloane's A013631). The positions at which thenumbers 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) |
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 is the Dirichlet Lambda Function. The above equations are special cases of a general result due toRamanujan  (Berndt 1985). Apéry's proof relied on showing that the sum
 | (7) |
 is a Binomial Coefficient, satisfies the Recurrence Relation
 | (8) |
Apéry's constant is also given by
 | (9) |
 is a Stirling Number of the First Kind. This can be rewritten as
 | (10) |
 is the  th Harmonic Number. Yet another expression for is
 | (11) |
Integrals for include
Gosper (1990) gave
 | (14) |
 | (15) |
 with
 | (16) |
 to obtain
 | (17) |
 ,
 | (18) |
 ,
 | (19) |
 for and 2 are
 | (20) |
 | (21) |
 | (22) |
 | (23) |
which gives 12 bits per term. The first few terms are
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-ZeilbergerPair identity with
 | (29) |
 , giving the rapidly converging
 | (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 Computationof Constants.'' http://www.lacim.uqam.ca/pi/records.html. Sloane, N. J. A.A013631,A033165, andA002117/M0020in ``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. |
