Apéry's constant

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In mathematics, at the crossing of number theory and special functions, Apéry's constant is defined as the number

\zeta(3) = \sum_{n=1}^\infty\frac{1}{n^3} = \lim_{n \to \infty}\left(\frac{1}{1^3} + \frac{1}{2^3} + \cdots + \frac{1}{n^3}\right)

where ζ is the Riemann zeta function. It has an approximate value of[1]

ζ(3) = 1.202056903159594285399738161511449990764986292...   (sequence A002117 in OEIS).
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 continued fraction is infinite, but it is not known whether this continued fraction is periodic or not.

This number 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.

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 (e.g., using Legendre polynomials). It is not known whether Apéry's constant is transcendental.

Since then, Wadim Zudilin and Tanguy Rivoal showed that infinitely many of the numbers ζ(2n+1) must be irrational,[2] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[3]

Series representation[edit]

In 1772, Leonhard Euler gave the series representation:[4]

\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.[5]

Ramanujan gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include:[6]

\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:[7]

\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[8] 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 a simple formula


\zeta(3) =\int\limits_0^1 \int\limits_0^1 \int\limits_0^1 \!  \frac{1}{1-xyz}\, dxdydz

which follows from summation representation for the Apéry's constant. Other simple formulas include


\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. More complicated representations are provided by Johan Jensen:[9]


\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

or F. Beukers:[10]


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

or Iaroslav Blagouchine:[11]


\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
.

Moreover, Evgrafov et al.'s 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)

is also very useful for the derivation of various integral representations via the known integral formulas for the Г- and polygamma-functions.[12]

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[13]
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[14]
April 2006 10,000,000,000 Shigeru Kondo & Steve Pagliarulo
January 2009 15,510,000,000 Alexander J. Yee & Raymond Chan[15]
March 2009 31,026,000,000 Alexander J. Yee & Raymond Chan[15]
September 2010 100,000,001,000 Alexander J. Yee[16]
September 2013 200,000,001,000 Robert J. Setti[16]
August 2015 250,000,000,000 Ron Watkins[16]

See also[edit]

Notes[edit]

  1. ^ See Wedeniwski 2001.
  2. ^ See Rivoal 2000.
  3. ^ See Zudilin 2001.
  4. ^ See Euler 1773.
  5. ^ See Srivastava 2000, p. 571 (1.11).
  6. ^ See Berndt 1989, chapter 14, formulas 25.1 and 25.3.
  7. ^ See Plouffe 1998.
  8. ^ See Broadhurst 1998.
  9. ^ See Jensen 1895.
  10. ^ See Beukers 1979.
  11. ^ See Blagouchine 2014.
  12. ^ See Evgrafov et al. 1969, exercise 30.10.1.
  13. ^ See Wedeniwski 2001.
  14. ^ See Gourdon & Sebah 2003.
  15. ^ a b See Yee 2009.
  16. ^ a b c See Yee 2015.

References[edit]

  • Apéry, Roger (1979), "Irrationalité de ζ(2) et ζ(3)", Astérisque 61: 11–13 .
  • Berndt, Bruce C. (1989), Ramanujan's notebooks, Part II, Springer .
  • Beukers, F. (1979), "A Note on the Irrationality of ζ(2) and ζ(3)", Bull. London Math. Soc. 11: 268–272 .
  • Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), arXiv:math.CA/9803067 .
  • Evgrafov, M. A.; Bezhanov, K. A.; Sidorov, Y. V.; Fedoriuk, M. V.; Shabunin, M. I. (1969), A Collection of Problems in the Theory of Analytic Functions [in Russian], Moscow: Nauka .
  • Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des mathématiciens (Gauthier-Villars) II: 346–347 .

Credits[edit]

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