Apéry's constant

In mathematics, at the intersection 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

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

Binary	1.0011001110111010…
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 the analysis of random minimum spanning trees and 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. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.

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

Series representations

Classical

In 1772, Leonhard Euler gave the series representation:

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

which was subsequently rediscovered several times.

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, then rediscovered and widely advertised by Apéry in 1979:

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

The following series representation, found by Amdeberhan in 1996, 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, 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, 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

$P(k)=126\,392k^{5}+412\,708k^{4}+531\,578k^{3}+336\,367k^{2}+104\,000k+12\,463.\,$

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

The following series representation, found by Mohamud Mohammed in 2005, 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

$P(k)=40\,885k^{5}+124\,346k^{4}+150\,160k^{3}+89\,888k^{2}+26\,629k+3116.\,$

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 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:

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

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

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

Srivastava 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:

$\zeta (3)=\int \limits _{0}^{1}\int \limits _{0}^{1}\int \limits _{0}^{1}\!{\frac {1}{1-xyz}}\,dx\,dy\,dz$.

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

$\zeta (3)={\frac {2}{3}}\int \limits _{0}^{\infty }\!{\frac {x^{2}}{e^{x}+1}}\,dx$.

This one follows from a Taylor expansion of $\chi _{3}(e^{ix})$ about $x=\pm \pi /2$, where $\chi _{\nu }(z)$ is the Legendre chi function:

$\zeta (3)={\frac {4}{7}}\int \limits _{0}^{\pi /2}x\log {(\sec {x}+\tan {x})}dx$

Note the similarity to

$G={\frac {1}{2}}\int \limits _{0}^{\pi /2}\log {(\sec {x}+\tan {x})}dx$

where $G$ is Catalan's constant.

More complicated formulas

For example, one formula was found by Johan Jensen:

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

another by F. Beukers:

$\zeta (3)=-{\frac {1}{2}}\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\ln(xy)}{\,1-xy\,}}\,dx\,dy$,

and yet another by Iaroslav Blagouchine:

$\zeta (3)=\,{\frac {8\pi ^{2}}{7}}\!\!\int \limits _{0}^{1}\!{\frac {x\left(x^{4}-4x^{2}+1\right)\ln \ln {\frac {1}{x}}}{\,(1+x^{2})^{4}\,}}\,dx\,=\,{\frac {8\pi ^{2}}{7}}\!\!\int \limits _{1}^{\infty }\!{\frac {x\left(x^{4}-4x^{2}+1\right)\ln \ln {x}}{\,(1+x^{2})^{4}\,}}\,dx$.

Evgrafov et al.'s connection to the derivatives of the gamma function

$\zeta (3)=-{\tfrac {1}{2}}\Gamma '''(1)+{\tfrac {3}{2}}\Gamma '(1)\Gamma ''(1)-\left(\Gamma '(1)\right)^{3}=-{\tfrac {1}{2}}\,\psi ^{(2)}(1)$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma-functions.

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
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
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
January 2009	15,510,000,000	Alexander J. Yee & Raymond Chan
March 2009	31,026,000,000	Alexander J. Yee & Raymond Chan
September 2010	100,000,001,000	Alexander J. Yee
September 2013	200,000,001,000	Robert J. Setti
August 2015	250,000,000,000	Ron Watkins
November 2015	400,000,000,000	Dipanjan Nag

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)

Main article: Particular values of Riemann zeta function

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. In 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.

See also

• Riemann zeta function
• Basel problem — ζ(2)
• List of sums of reciprocals

References

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• Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", El. J. Combinat., 4 (2).
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• Berndt, Bruce C. (1989), Ramanujan's notebooks, Part II, Springer.
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