Apéry's constant explained

Rationality:	Irrational
Symbol:	ζ(3)

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

\begin{align} \zeta( 3 ) & = \sum ^{inf ty} _{n =1} (1)/(n ^{3}) \\ & = \lim _{n} \left( (1)/(1 ^{3}) + (1)/(2 ^{3}) + … + (1)/(n ^{3}) \right), \end{align}

where is the Riemann zeta function. It has an approximate value of

.

It is named after Roger Apéry, who proved that it is an irrational number.

Apéry's 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

See main article: Apéry's theorem. was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.^[1]

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for,

\zeta( 3 ) = \int ^{1} _{0} \int ^{1} _{0} \int ^{1} _{0} (1)/(1- xyz) dx dy dz, by the Legendre polynomials.In particular, van der Poorten's article chronicles this approach by noting thatI _{3} : =- (1)/(2) \int ^{1} _{0} \int ^{1} _{0} (P _{n(x)} P _{n(y)} log (xy))/(1- xy) dx dy = b _{n} \zeta( 3 ) - a _{n,}

where

|I| \leq \zeta( 3 ) ( 1- \sqrt{ 2 }) ^{4n} , P _{n(z)} are the Legendre polynomials, and the subsequences b _{n,} 2 \operatorname{lcm}( 1 , 2 ,\ldots,n) ⋅ a _{n} \in 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:

\zeta( 3 ) = \sum ^{inf ty} _{k =1} (1)/(k ^{3}) , Leonhard Euler gave the series representation:\zeta( 3 ) = (\pi ^{2})/(7) \left( 1 - 4 \sum ^{inf ty} _{k =1} (\zeta ( 2 k))/(2 ^{2k} ( 2 k +1 )( 2 k +2 )) \right) in 1772, which was subsequently rediscovered several times.

Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of . 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, rediscovered by Hjortnaes in 1953, and rediscovered once more and widely advertised by Apéry in 1979:

\zeta( 3 ) = (5)/(2) \sum ^{inf ty} _{k =1} ( -1 ) ^{k-1} (k! ^{2})/(( 2 k)! k ^{3}) .

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

\zeta( 3 ) = (1)/(4) \sum ^{inf ty} _{k =1} ( -1 ) ^{k-1} ((k -1 )! ^{3} ( 56 k ^{2} -32 k +5 ))/(( 2 k -1 ) ^{2(3k)!}) .

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

\zeta( 3 ) = (1)/(64) \sum ^{inf ty} _{k =0} ( -1 ) ^{k} (k! ^{10} ( 205 k ^{2} +250 k +77 ))/(( 2 k +1 )! ^{5}) .

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:^[2]

\zeta( 3 ) = (1)/(24) \sum ^{inf ty} _{k =0} ( -1 ) ^{k} (( 2 k +1 )! ^{3} ( 2 k)! ^{3} k! ^{3} ( 126392 k ^{5} +412708 k ^{4} +531578 k ^{3} +336367 k ^{2} +104000 k +12463 ))/(( 3 k +2 )! ( 4 k +3 )! ^{3}) . It has been used to calculate Apéry's constant with several million correct decimal places.^[3]

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

\zeta( 3 ) = (1)/(2) \sum ^{inf ty} _{k =0} (( -1 ) ^{k} ( 2 k)! ^{3} (k +1 )! ^{6} ( 40885 k ^{5} +124346 k ^{4} +150160 k ^{3} +89888 k ^{2} +26629 k +3116 ))/((k +1 ) ^{2} ( 3 k +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.

Thue-Morse sequence

The following representation was found by Tóth in 2022:

\begin{align} \sum _{n\geq1} (9 t _{n-1} +7 t _{n})/(n ^{3}) & = 8 \zeta( 3 ),\end{align} where (t _{n)} _{n\geq0} is the n ^{\rm} term of the Thue-Morse sequence. In fact, this is a special case of the following formula (valid for all s with real part greater than 1 ):( 2 ^{s+1)} \sum _{n\geq1} (t _{n-1})/(n ^{s}) + ( 2 ^{s-1)} \sum _{n\geq1} (t _{n})/(n ^{s}) = 2 ^{s} \zeta(s).

Others

The following series representation was found by Ramanujan:^[4]

\zeta( 3 ) = (7)/(180) \pi ^{3} - 2 \sum ^{inf ty} _{k =1} (1)/(k ^{3} (e ^{2\pi} -1 )) .

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

\zeta( 3 ) = 14 \sum ^{inf ty} _{k =1} (1)/(k ^{3} \sinh(\pi k)) - (11)/(2) \sum ^{inf ty} _{k =1} (1)/(k ^{3} (e ^{2\pi} -1 )) - (7)/(2) \sum ^{inf ty} _{k =1} (1)/(k ^{3} (e ^{2\pi} +1 )) .

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

The following formula follows directly from the integral definition of the zeta function:

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

More complicated formulas

Other formulas include

\zeta( 3 ) = \pi \int ^{inf ty} _{0} (\cos( 2 \arctan x))/((x ^{2} + 1 ) \left(\cosh (1)/(2) \pi x\right) ^{2}) dx and\zeta( 3 ) =- (1)/(2) \int ^{1} _{0} \int ^{1} _{0} (log (xy))/(1- xy) dxdy = - \int ^{1} _{0} \int ^{1} _{0} (log ( 1- xy))/(xy) dxdy.

Also,

\begin{align} \zeta( 3 ) & = (8 \pi ^{2})/(7) \int ^{1} _{0} (x(x ^{4} - 4 x ^{2} + 1 ) loglog (1)/(x))/(( 1+ x ^{2)} ^{4}) dx \\ & = (8 \pi ^{2})/(7) \int ^{inf ty} _{1} (x(x ^{4} -4 x ^{2} +1 ) loglog {x)/() } \,dx. \end

A connection to the derivatives of the gamma function

\zeta( 3 ) = - \tfrac{ 1 }{ 2 }(\Gamma'''( 1 ) + \gamma ^{3+} \tfrac{ 1 }{ 2 }\pi ^{2\gamma)} = - \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 has increased dramatically during the last decades, and now stands at more than . This is due both to the increasing performance of computers and to algorithmic improvements.

Decimal digits	Computation performed by
1735	16	Leonhard Euler
Unknown	16	Adrien-Marie Legendre
1887	32	Thomas Joannes Stieltjes
1996		Greg J. Fee & Simon Plouffe
1997		Bruno Haible & Thomas Papanikolaou
May 1997		Patrick Demichel
February 1998		Sebastian Wedeniwski
March 1998		Sebastian Wedeniwski
July 1998		Sebastian Wedeniwski
December 1998		Sebastian Wedeniwski
September 2001		Shigeru Kondo & Xavier Gourdon
February 2002		Shigeru Kondo & Xavier Gourdon
February 2003		Patrick Demichel & Xavier Gourdon
April 2006		Shigeru Kondo & Steve Pagliarulo
January 21, 2009		Alexander J. Yee & Raymond Chan
February 15, 2009		Alexander J. Yee & Raymond Chan
September 17, 2010		Alexander J. Yee
September 23, 2013		Robert J. Setti
August 7, 2015		Ron Watkins
December 21, 2015		Dipanjan Nag
August 13, 2017		Ron Watkins
May 26, 2019		Ian Cutress^[5]
July 26, 2020		Seungmin Kim^[6]
December 22, 2023		Andrew Sun

Reciprocal

The reciprocal of (0.8319073725807...) is the probability that any three positive integers, chosen at random, will be relatively prime, in the sense that as approaches infinity, the probability that three positive integers less than chosen uniformly at random will not share a common prime factor approaches this value. (The probability for n positive integers is .) In the same sense, it is the probability that a positive integer chosen at random will not be evenly divisible by the cube of an integer greater than one. (The probability for not having divisibility by an n-th power is .)

Extension to

See main article: Particular values of the Riemann zeta function. Many people have tried to extend Apéry's proof that is irrational to other values of the zeta function with odd arguments. Infinitely many of the numbers must be irrational, and at least one of the numbers,,, and must be irrational.

See also

• Riemann zeta function
• Basel problem —
• List of sums of reciprocals

References

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• (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).
• (Message to Simon Plouffe, with original text but only some decimal places).
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Further reading

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External links

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Notes and References

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2. . In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from . The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).

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