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

$$\zeta(3)=1.2020569\ldots,$$ (1)

(Sloane's A002117) where $\zeta(z)$ is the Riemann Zeta Function. Apéry (1979) proved that $\zeta(3)$ is Irrational, although it is not known if it is Transcendental. The Continued Fraction for $\zeta(3)$ 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 $\zeta(3)$ are

$$\zeta(3)={5\over 2}\sum_{n=1}^\infty {(-1)^{n-1}\over n^3{2n... ... {5\over 2}\sum_{k=1}^\infty {(-1)^{k+1}(k!)^2\over (2k)! k^3}$$ (2)

(used by Apéry), and

$$\lambda(3)=\sum_{k=0}^\infty {1\over (2k+1)^3}={\textstyle{7\over 8}}\zeta(3)$$ (3)

$$\sum_{k=0}^\infty {1\over (3k+1)^3} = {2\pi^3\over 81\sqrt{3}} +{\textstyle{13\over 27}}\zeta(3)$$ (4)

$$\sum_{k=0}^\infty {1\over (4k+1)^3} = {\pi^3\over 64}+{\textstyle{7\over 16}}\zeta(3)$$ (5)

$$\sum_{k=0}^\infty {1\over (6k+1)^3} = {\pi^3\over 36\sqrt{3}}+{\textstyle{91\over 216}}\zeta(3),$$ (6)

where $\lambda(z)$ 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

$$a(n)\equiv \sum_{k=0}^n{n\choose k}^2{n+k\choose k}^2,$$ (7)

where ${n\choose k}$ is a Binomial Coefficient, satisfies the Recurrence Relation

$$(n+1)^3a(n+1)-(34n^3+51n^2+27n+5)a(n)+n^3a(n-1)=0$$ (8)

(van der Poorten 1979, Zeilberger 1991).

Apéry's constant is also given by

$$\zeta(3)=\sum_{n=1}^\infty {S_{n,2}\over n! n},$$ (9)

where $S_{n,m}$ is a Stirling Number of the First Kind. This can be rewritten as

$$\sum_{n=1}^\infty {H_n\over n^2}=2\zeta(3),$$ (10)

where $H_n$ is the $n$th Harmonic Number. Yet another expression for $\zeta(3)$ is

$$\zeta(3)={1\over 2}\sum_{n=1}^\infty {1\over n^2}\left({1+{1\over 2}+\ldots+{1\over n}}\right)$$ (11)

(Castellanos 1988).

Integrals for $\zeta(3)$ include

$$\zeta(3) = {1\over 2}\int_0^\infty {t^2\over e^t-1}\,dt$$ (12)

$$= {8\over 7}\left[{{\textstyle{1\over 4}}\pi^2\ln 2+2\int_0^{\pi/4} x\ln(\sin x)\,dx}\right].$$ (13)

Gosper (1990) gave

$$\zeta(3)={1\over 4}\sum_{k=1}^\infty {30k-11\over (2k-1)k^3{2k\choose k}^2}.$$ (14)

A Continued Fraction involving Apéry's constant is

$${6\over\zeta(3)}=5-{1^6\over 117-} {2^6\over 535-} \cdots {n^6\over 34n^3+51n^2+27n+5-} \cdots$$ (15)

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

$$F(n,k)={(-1)^kk!^2(sn-k-1)!\over (sn+k+1)!(k+1)},$$ (16)

$s=1$ to obtain

$$\zeta(3)={5\over 2}\sum_{n=1}^\infty (-1)^{n-1} {1\over{2n\choose n}n^3}.$$ (17)

For $s=2$,

$$\zeta(3)={1\over 4}\sum_{n=1}^\infty (-1)^{n-1} {56n^2-32+5\over (2n-1)^2}{1\over{3n\choose n}{2n\choose n}n^3}$$ (18)

and for $s=3$,

$$\zeta(3)=\sum_{n=0}^\infty {(-1)^n\over 72{4n\choose n}{3n\c... ...+13761n^2+13878n^3+1040\over(4n+1)(4n+3)(n+1)(3n+1)^2(3n+2)^2}$$ (19)

(Amdeberhan 1996). The corresponding $G(n,k)$ for $s=1$ and 2 are

$$G(n,k)={2(-1)^k k!^2(n-k)!\over(n+k+1)!(n+1)^2}$$ (20)

and

$$G(n,k)={(-1)^kk!^2(2n-k)!(3+4n)(4n^2+6n+k+3)\over 2(2n+k+2)!(n+1)^2(2n+1)^2}.$$ (21)

Gosper (1996) expressed $\zeta(3)$ as the Matrix Product

$$\lim_{N\to\infty} \prod_{n=1}^N M_n = \left[{\matrix{0 & \zeta(3)\cr 0 & 1\cr}}\right],$$ (22)

where

$$M_n\equiv\left[{\matrix{ {(n+1)^4\over 4096(n+{\textstyle{5\... ...le{1\over 2}})(n+{\textstyle{2\over 3}})}\cr 0 & 1\cr}}\right]$$ (23)

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

$$M_1 = \left[\begin{array}{cc}{\textstyle{1\over 19600}} & {\textstyle{2077\over 1728}}\\ 0 & 1\end{array}\right]$$ (24)

$$M_2 = \left[\begin{array}{cc}{\textstyle{1\over 9801}} & {\textstyle{7561\over 4320}}\\ 0 & 1\end{array}\right]$$ (25)

$$M_3 = \left[\begin{array}{cc}{\textstyle{9\over 67600}} & {\textstyle{50501\over 20160}}\\ 0 & 1\end{array}\right],$$ (26)

which gives

$$\zeta(3)\approx {\textstyle{423203577229\over 352066176000}} = 1.20205690315732\ldots.$$ (27)

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

$$[\zeta(3)]^{-1}\approx 1.20206^{-1} \approx 0.831907.$$ (28)

B. Haible and T. Papanikolaou computed $\zeta(3)$ to 1,000,000 Digits using a Wilf-Zeilberger Pair identity with

$$F(n,k)=(-1)^k {n!^6(2n-k-1)!k!^3\over 2(n+k+1)!^2(2n)!^3},$$ (29)

$s=1$, and $t=1$, giving the rapidly converging

$$\zeta(3)=\sum_{n=0}^\infty (-1)^n{n!^{10}(205n^2+250n+77)\over 64(2n+1)!^5}$$ (30)

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

See also Riemann Zeta Function, Wilf-Zeilberger Pair

References

Amdeberhan, T. ``Faster and Faster Convergent Series for $\zeta(3)$.'' 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 $\zeta(2)$ et $\zeta(3)$.'' 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 $\zeta(3)$.'' 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 $\zeta(3)$.'' 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 $\zeta(3)$.'' Math. Intel. 1, 196-203, 1979.

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