Apéry's constant

In mathematics, at the intersection of number theory and special functions, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

\( {\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}} \)
where ζ is the Riemann zeta function. It has an approximate value of[1]

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

The constant is named after Roger Apéry. 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 the analysis of random minimum spanning trees[2] 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 after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.[3] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[4] and simpler proofs were found later.[5]

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for \( \zeta (3) \) ,

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

by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that

\( {\displaystyle I_{3}:=-{\frac {1}{2}}\int _{0}^{1}\int _{0}^{1}{\frac {P_{n}(x)P_{n}(y)\log(xy)}{1-xy}}\,dx\,dy=b_{n}\zeta (3)-a_{n},} \)

where \( {\displaystyle |I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}} \), \( P_{n}(z) \) are the Legendre polynomials, and the subsequences \( {\displaystyle b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z} } \) are integers or almost integers.

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

In addition to the fundamental series:

\( {\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},} \)

Leonhard Euler gave the series representation:[6]

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

in 1772, which was subsequently rediscovered several times.[7]

Other classical series representations include:

\( {\displaystyle {\begin{aligned}\zeta (3)&={\frac {8}{7}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)^{3}}}\\\zeta (3)&={\frac {4}{3}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(k+1)^{3}}}\end{aligned}}} \)

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 A. A. Markov in 1890,[8] rediscovered by Hjortnaes in 1953,[9] and rediscovered once more and widely advertised by Apéry in 1979:[3]

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

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[10]

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

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[11]

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

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[12]

\( {\displaystyle \zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}} \)

It has been used to calculate Apéry's constant with several million correct decimal places.[13]

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[14]

\( {\displaystyle \zeta (3)={\frac {1}{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}} \)

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.[15]
Others

The following series representation was found by Ramanujan:[16]

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

The following series representation was found by Simon Plouffe in 1998:[17]

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

Srivastava (2000) 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:

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

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

\( {\displaystyle \zeta (3)={\frac {1}{2}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx} \)

and

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

This one follows from a Taylor expansion of χ3(eix) about x = ±π/2, where χν(z) is the Legendre chi function:

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

Note the similarity to

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

where G is Catalan's constant.
More complicated formulas

Other formulas include:[18]

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

and,[19]

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

Mixing these two formulas, one can obtain :

\( {\displaystyle \zeta (3)=\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,x\,}}\,dx}, \)

By symmetry,

\( {\displaystyle \zeta (3)=\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,1-x\,}}\,dx}, \)

Summing both, \( {\displaystyle \zeta (3)={\frac {1}{2}}\int _{0}^{1}\!\!{\frac {\log(x)\log(1-x)}{\,x(1-x)\,}}\,dx}. \)

Also,[20]

\( {\displaystyle {\begin{aligned}\zeta (3)&={\frac {8\pi ^{2}}{7}}\!\!\int _{0}^{1}\!{\frac {x\left(x^{4}-4x^{2}+1\right)\log \log {\frac {1}{x}}}{\,(1+x^{2})^{4}\,}}\,dx\\&={\frac {8\pi ^{2}}{7}}\!\!\int _{1}^{\infty }\!{\frac {x\left(x^{4}-4x^{2}+1\right)\log \log {x}}{\,(1+x^{2})^{4}\,}}\,dx\end{aligned}}}. \)

A connection to the derivatives of the gamma function

\( {\displaystyle \zeta (3)=-{\tfrac {1}{2}}\Gamma '''(1)+{\tfrac {3}{2}}\Gamma '(1)\Gamma ''(1)-{\big (}\Gamma '(1){\big )}^{3}=-{\tfrac {1}{2}}\,\psi ^{(2)}(1)} \)

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma-functions.[21]
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	520000	Greg J. Fee & Simon Plouffe
1997	1000000	Bruno Haible & Thomas Papanikolaou
May 1997	10536006	Patrick Demichel
February 1998	14000074	Sebastian Wedeniwski
March 1998	32000213	Sebastian Wedeniwski
July 1998	64000091	Sebastian Wedeniwski
December 1998	128000026	Sebastian Wedeniwski^[1]
September 2001	200001000	Shigeru Kondo & Xavier Gourdon
February 2002	600001000	Shigeru Kondo & Xavier Gourdon
February 2003	1000000000	Patrick Demichel & Xavier Gourdon^[22]
April 2006	10000000000	Shigeru Kondo & Steve Pagliarulo
January 21, 2009	15510000000	Alexander J. Yee & Raymond Chan^[23]
February 15, 2009	31026000000	Alexander J. Yee & Raymond Chan^[23]
September 17, 2010	100000001000	Alexander J. Yee^[24]
September 23, 2013	200000001000	Robert J. Setti^[24]
August 7, 2015	250000000000	Ron Watkins^[24]
December 21, 2015	400000000000	Dipanjan Nag^[25]
August 13, 2017	500000000000	Ron Watkins^[24]
May 26, 2019	1000000000000	Ian Cutress^[26]
July 26, 2020	1200000000100	Seungmin Kim^[27]^[28]

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).[29]
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 values of the zeta function with odd arguments. Infinitely many of the numbers ζ(2n + 1) must be irrational,[30] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[31]
See also

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

Notes

Wedeniwski (2001).
Frieze (1985).
Apéry (1979).
van der Poorten (1979).
Beukers (1979); Zudilin (2002).
Euler (1773).
Srivastava (2000), p. 571 (1.11).
Markov (1890).
Hjortnaes (1953).
Amdeberhan (1996).
Amdeberhan & Zeilberger (1997).
Wedeniwski (1998); 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).
Wedeniwski (1998); Wedeniwski (2001).
Mohammed (2005).
Broadhurst (1998).
Berndt (1989, chapter 14, formulas 25.1 and 25.3).
Plouffe (1998).
Jensen (1895).
Beukers (1979).
Blagouchine (2014).
Evgrafov et al. (1969), exercise 30.10.1.
Gourdon & Sebah (2003).
Yee (2009).
Yee (2017).
Nag (2015).
Records set by y-cruncher, retrieved June 8, 2019
Records set by y-cruncher, archived from the original on 2020-08-10, retrieved August 10, 2020
Apéry's constant world record by Seungmin Kim, retrieved July 28, 2020
Mollin (2009).
Rivoal (2000).

Zudilin (2001).

References

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Apéry, Roger (1979), "Irrationalité de ζ 2 {\displaystyle \zeta 2} {\displaystyle \zeta 2} et ζ 3 {\displaystyle \zeta 3} {\displaystyle \zeta 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 ) {\displaystyle \zeta (2)} \zeta (2) and ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3)", Bull. London Math. Soc., 11 (3): 268–272, doi:10.1112/blms/11.3.268.
Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results", The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5, S2CID 120943474.
Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ ( 3 ) {\displaystyle \zeta (3)} \zeta (3) and ζ ( 5 ) {\displaystyle \zeta (5)} \zeta (5), arXiv:math.CA/9803067.
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Nag, Dipanjan (2015), Calculated Apéry's constant to 400,000,000,000 Digit, A world record
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