# Apéry's constant

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).
 γ ζ(3) √2 √3 √5 φ ρ δS e π δ 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 constant 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.

## Irrational number

ζ(3) was named Apéry's constant for the French mathematician Roger Apéry, who proved in 1978 that it is irrational.[2] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[3] and simpler proofs were found later.[4][5]

It is still not known whether Apéry's constant is transcendental.

## Series representations

### Classical

In 1772, Leonhard Euler gave the series representation:[6]

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

which was subsequently rediscovered several times.[7]

Other classical series representations include:

$\zeta(3) = \frac{8}{7} \sum_{k=0}^\infty \frac{1}{(2k+1)^3}$
$\zeta(3) = \frac{4}{3} \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)^3}$

### Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of ζ(3). Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "Known digits").

The following series representation was found by Hjortnaes in 1953,[8] then rediscovered and widely advertised by Apéry in 1979:[2]

$\zeta(3) = \frac{5}{2} \sum_{k=1}^\infty (-1)^{k-1} \frac{k!^2}{k^3 (2k)!} = \frac{5}{2} \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^3 \tbinom{2k}{k}}$

The following series representation, found by Amdeberhan in 1996,[9] gives (asymptotically) 1.43 new correct decimal places per term:

$\zeta(3) = \frac{1}{4} \sum_{k=1}^\infty (-1)^{k-1} \frac{56k^2 - 32k + 5}{(2k-1)^2} \frac{(k-1)!^3}{(3k)!}$

The following series representation, found by Amdeberhan and Zeilberger in 1997,[10] gives (asymptotically) 3.01 new correct decimal places per term:

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

The following series representation, found by Sebastian Wedeniwski in 1998,[11] gives (asymptotically) 5.04 new correct decimal places per term:

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

where

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

It was used used by Wedeniwski to calculate Apéry's constant with several million correct decimal places.[12]

The following series representation, found by Mohamud Mohammed in 2005,[13] gives (asymptotically) 3.92 new correct decimal places per term:

$\zeta(3) = \frac{1}{2}\,\sum_{k=0}^\infty \frac{(-1)^k(2k)!^3(k+1)!^6} {(k+1)!^2(3k+3)!^4}\,P(k)$

where

$P(k) = 40885k^5 + 124346k^4 + 150160k^3 + 89888k^2 + 26629k + 3116.\,$

The following series representation has an undetermined source:

$\zeta(3) = \frac{8}{7}-\frac{8}{7}\sum_{k=1}^\infty \frac{{\left( -1 \right) }^k\,2^{12k-5}\,k\, \left( 7168k^5 - 2688k^4 - 432k^3 + 148k^2 + 9k - 3 \right) \, {k!}^3\,{\left( 2k-1 \right) !}^6}{{\left( 2k-1 \right) }^3\, \left( 3k \right) !\,{\left( 4k+1 \right) !}^3}.$

### Digit by digit

In 1998, Broadhurst[14] gave 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.

### Others

The following series representation was found by Ramanujan:[15]

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

The following series representation was found by Simon Plouffe in 1998:[16]

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

Srivastava[17] collected many series that converge to Apéry's constant.

## Integral representations

There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

### Simple formulas

For example, this one follows from the summation representation for Apéry's constant:

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

The next two follow directly from the well-known integral formulas for the Riemann zeta function:

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

and

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

### More complicated formulas

For example, one formula was found by Johan Jensen:[18]

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

another by F. Beukers:[4]

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

and yet another by Iaroslav Blagouchine:[19]

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

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.[20]

## Known digits

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increasing 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
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[1]
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[21]
April 2006 10,000,000,000 Shigeru Kondo & Steve Pagliarulo
January 2009 15,510,000,000 Alexander J. Yee & Raymond Chan[22]
March 2009 31,026,000,000 Alexander J. Yee & Raymond Chan[22]
September 2010 100,000,001,000 Alexander J. Yee[23]
September 2013 200,000,001,000 Robert J. Setti[23]
August 2015 250,000,000,000 Ron Watkins[23]

## Reciprocal

The reciprocal of ζ(3) 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).

## Extension to ζ(2n+1)

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other odd zeta values. In 2000, Tanguy Rivoal showed that infinitely many of the numbers ζ(2n+1) must be irrational.[24] In 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[25]

## Notes

1. ^ a b See Wedeniwski 2001.
2. ^ a b See Apéry 1979.
3. ^
4. ^ a b See Beukers 1979.
5. ^ See Zudilin 2002.
6. ^ See Euler 1773.
7. ^ See Srivastava 2000, p. 571 (1.11).
8. ^ See Hjortnaes 1953.
9. ^ See Amdeberhan 1996.
10. ^
11. ^ See Wedeniwski 1998 and Wedeniwski 2001. In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from Amdeberhan & Zeilberger 1997. The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).
12. ^
13. ^ See Mohammed 2005.
15. ^ See Berndt 1989, chapter 14, formulas 25.1 and 25.3.
16. ^ See Plouffe 1998.
17. ^ See Srivastava 2000.
18. ^ See Jensen 1895.
19. ^ See Blagouchine 2014.
20. ^ See Evgrafov et al. 1969, exercise 30.10.1.
21. ^
22. ^ a b See Yee 2009.
23. ^ a b c See Yee 2015.
24. ^ See Rivoal 2000.
25. ^ See Zudilin 2001.

## References

• 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.
• Hjortnaes, M. M. (August 1953), Overføring av rekken $\sum_{k=1}^\infty(\frac{1}{k^3})$ til et bestemt integral, in Proc. 12th Scandinavian Mathematical Congress, Lund, Sweden: Scandinavian Mathematical Society, pp. 211–213 .
• 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.
• Mohammed, Mohamud (2005), "Infinite families of accelerated series for some classical constants by the Markov-WZ method", Discrete Mathematics and Theoretical Computer Science (Springer) 7: 11–24.
• Wedeniwski, Sebastian (2001), Simon Plouffe, ed., The Value of Zeta(3) to 1,000,000 places, Project Gutenberg (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).