Apéry's constant

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Binary 1.001100111011101...
Decimal 1.2020569031595942854...
Hexadecimal 1.33BA004F00621383...
Continued fraction 1 + \frac{1}{4 + \cfrac{1}{1 + \cfrac{1}{18 + \cfrac{1}{\ddots\qquad{}}}}}
Note that this continuing fraction is not periodic.

In mathematics, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law. It is defined as the number ζ(3),

\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}=1+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \frac{1}{6^3} + \frac{1}{7^3} + \frac{1}{8^3} + \frac{1}{9^3} + \cdots\,\!

where ζ is the Riemann zeta function. It has an approximate value of (Wedeniwski 2001)

ζ(3) = 1.202056903159594285399738161511449990764986292...   (sequence A002117 in OEIS).

The reciprocal of this constant is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as N goes to infinity, the probability that three positive integers less than N chosen uniformly at random will be relatively prime approaches this value).

Apéry's theorem[edit]

Main article: Apéry's theorem

This value was named for Roger Apéry (1916–1994), who in 1978 proved it to be irrational. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and shorter proofs have been found later, using Legendre polynomials. It is not known whether Apéry's constant is transcendental.

Work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2n+1) must be irrational,[1] and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[2]

Series representation[edit]

In 1772, Leonhard Euler (Euler 1773) gave the series representation (Srivastava 2000, p. 571 (1.11)):

\zeta(3)=\frac{\pi^2}{7}
\left[ 1-4\sum_{k=1}^\infty \frac {\zeta (2k)} {(2k+1)(2k+2) 2^{2k}} \right]

which was subsequently rediscovered several times.

Ramanujan gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include:[3]

\zeta(3)=\frac{7}{180}\pi^3 -2 
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}

Simon Plouffe has developed other series (Plouffe 1998):

\zeta(3)= 14 
\sum_{k=1}^\infty \frac{1}{k^3 \sinh(\pi k)}
-\frac{11}{2}
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} -1)}
-\frac{7}{2} 
\sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} +1)}.

Similar relations for the values of \zeta(2n+1) are given in the article zeta constants.

Many additional series representations have been found, including:

\zeta(3) = \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k+1)^3}
\zeta(3) = \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^3}
\zeta(3) = \frac{5}{2} \sum_{k=1}^\infty (-1)^{k-1} \frac{k!^2}{k^3 (2k)!}
\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1}
\frac{56k^2-32k+5}{(2k-1)^2} \frac{(k-1)!^3}{(3k)!}
\zeta(3)=\frac{8}{7}-\frac{8}{7}\sum_{k=1}^\infty \frac{{\left( -1 \right) }^k\,2^{-5 + 12\,k}\,k\,
    \left( -3 + 9\,k + 148\,k^2 - 432\,k^3 - 2688\,k^4 + 7168\,k^5 \right) \,
    {k!}^3\,{\left( -1 + 2\,k \right) !}^6}{{\left( -1 + 2\,k \right) }^3\,
    \left( 3\,k \right) !\,{\left( 1 + 4\,k \right) !}^3}
\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{205k^2 + 250k + 77}{64} \frac{k!^{10}}{(2k+1)!^5}

and

\zeta(3) = \sum_{k=0}^\infty (-1)^k \frac{P(k)}{24}
\frac{((2k+1)!(2k)!k!)^3}{(3k+2)!(4k+3)!^3}

where

P(k) = 126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463.\,

Some of these have been used to calculate Apéry's constant with several million digits.

(Broadhurst 1998) gives a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.

Integral representations[edit]

There are also numerous integral representations for the Apéry's constant. These include simple formulas, such as


\zeta(3) =\frac{1}{2}\int\limits_0^\infty \!  \frac{x^2}{e^x-1}\, dx

or


\zeta(3) =\frac{2}{3}\int\limits_0^\infty \!  \frac{x^2}{e^x+1}\, dx

which follow directly from the well-known integral formulas for the Riemann zeta function, as well as more complicated representations such as


\zeta(3)=\pi\!\!\int\limits_{0}^{\infty} \! 
\frac{\cos(2\arctan\,x)}{\left(x^2+1\right)\big[\cosh\frac{1}{2}\pi x\big]^2}\, dx

see Johan Jensen,[4] or


\zeta(3) =-\frac{1}{2}\int\limits_0^1 \!\!\int\limits_0^1  \frac{\ln(xy)}{\,1-xy\,}\, dx \, dy

see F. Beukers,[5] or


\zeta(3) =\,\frac{8\pi^2}{7}\!\!\int\limits_0^1 \!  \frac{x\left(x^4-4x^2+1\right)\ln\ln\frac{1}{x}}{\,(1+x^2)^4\,}\, dx \,=\,
\frac{8\pi^2}{7}\!\!\int\limits_1^\infty \!\frac{x\left(x^4-4x^2+1\right)\ln\ln{x}}{\,(1+x^2)^4\,}\, dx

see Iaroslav Blagouchine.[6] Moreover, the connection to the derivatives of the Г-function


\zeta(3) = -\frac{1}{2}\Gamma'''(1)+\frac{3}{2}\Gamma'(1)\Gamma''(1)- [\Gamma'(1)]^3 = -\frac{1}{2} \, \psi^{(2)}(1)

see., e.g., exercise 30.10.1 in,[7] is also very useful for the derivation of various integral representations via the known integral formulas for the Г- and polygamma-functions.

Known digits[edit]

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increase of performance of computers and to algorithmic improvements.

Number of known decimal digits of Apéry's constant ζ(3)
Date Decimal digits Computation performed by
1735 16 Leonhard Euler
unknown 16 Adrien-Marie Legendre
1887 32 Thomas Joannes Stieltjes
1996 520,000 Greg J. Fee & Simon Plouffe
1997 1,000,000 Bruno Haible & Thomas Papanikolaou
May 1997 10,536,006 Patrick Demichel
February 1998 14,000,074 Sebastian Wedeniwski
March 1998 32,000,213 Sebastian Wedeniwski
July 1998 64,000,091 Sebastian Wedeniwski
December 1998 128,000,026 Sebastian Wedeniwski (Wedeniwski 2001)
September 2001 200,001,000 Shigeru Kondo & Xavier Gourdon
February 2002 600,001,000 Shigeru Kondo & Xavier Gourdon
February 2003 1,000,000,000 Patrick Demichel & Xavier Gourdon
April 2006 10,000,000,000 Shigeru Kondo & Steve Pagliarulo (see Gourdon & Sebah (2003))
January 2009 15,510,000,000 Alexander J. Yee & Raymond Chan (see Yee & Chan (2009))
March 2009 31,026,000,000 Alexander J. Yee & Raymond Chan (see Yee & Chan (2009))
September 2010 100,000,001,000 Alexander J. Yee (see Yee)
September 2013 200,000,001,000 Robert J. Setti (Apéry's Constant - Zeta(3) - 200 Billion Digits)

Notes[edit]

  1. ^ T. Rivoal (2000), "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 331 (4): 267–270, Bibcode:2000CRASM.331..267R, doi:10.1016/S0764-4442(00)01624-4. 
  2. ^ W. Zudilin (2001), "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational", Russ. Math. Surv. 56 (4): 774–776, Bibcode:2001RuMaS..56..774Z, doi:10.1070/RM2001v056n04ABEH000427. 
  3. ^ Bruce C. Berndt, Ramanujan's notebooks, Part II (1989), Springer-Verlag. See chapter 14, formulas 25.1 and 25.3
  4. ^ Johan Ludwig William Valdemar Jensen. Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver. L'Intermédiaire des mathématiciens, tome II, pp. 346-347, 1895.
  5. ^ F. Beukers A Note on the Irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11, pp. 268-272, 1979.
  6. ^ Iaroslav V. Blagouchine Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. PDF
  7. ^ M. A. Evgrafov and K. A. Bezhanov and Y. V. Sidorov and M. V. Fedoriuk and M. I. Shabunin. A Collection of Problems in the Theory of Analytic Functions [in Russian]. Nauka, Moscow, 1969.

References[edit]

This article incorporates material from Apéry's constant on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.