Apéry's constant

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  • List of numbers

  • Irrational numbers




  • ζ(3)

  • √2

  • √3

  • √5

  • φ

  • e

  • π



Binary

1.0011001110111010…

Decimal

1.2020569031595942854…

Hexadecimal

1.33BA004F00621383…

Continued fraction

1+14+11+118+1⋱displaystyle 1+frac 14+cfrac 11+cfrac 118+cfrac 1ddots qquad 1 + frac14 + cfrac11 + cfrac118 + cfrac1ddotsqquad
Note that this continued fraction is infinite, but it is not known whether this continued fraction is periodic or not.

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


ζ(3)=∑n=1∞1n3=limn→∞(113+123+⋯+1n3)displaystyle beginalignedzeta (3)&=sum _n=1^infty frac 1n^3\&=lim _nto infty left(frac 11^3+frac 12^3+cdots +frac 1n^3right)endaligneddisplaystyle beginalignedzeta (3)&=sum _n=1^infty frac 1n^3\&=lim _nto infty left(frac 11^3+frac 12^3+cdots +frac 1n^3right)endaligned

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



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

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[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.




Contents





  • 1 Irrational number


  • 2 Series representations

    • 2.1 Classical


    • 2.2 Fast convergence


    • 2.3 Digit by digit


    • 2.4 Others



  • 3 Integral representations

    • 3.1 Simple formulas


    • 3.2 More complicated formulas



  • 4 Known digits


  • 5 Reciprocal


  • 6 Extension to ζ(2n + 1)


  • 7 See also


  • 8 Notes


  • 9 References


  • 10 Credits




Irrational number


ζ(3) was named Apéry's constant for the French mathematician Roger Apéry, who proved in 1978 that it is irrational.[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][6]


Beuker's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3)displaystyle zeta (3)zeta (3),


ζ(3)=∫01∫01∫0111−xyzdxdydz,displaystyle zeta (3)=int _0^1int _0^1int _0^1frac 11-xyzdx,dy,dz,displaystyle zeta (3)=int _0^1int _0^1int _0^1frac 11-xyzdx,dy,dz,

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


I3:=−12∫01∫01Pn(x)Pn(y)log⁡(xy)1−xydxdy=bnζ(3)−an,displaystyle I_3:=-frac 12int _0^1int _0^1frac P_n(x)P_n(y)log(xy)1-xydx,dy=b_nzeta (3)-a_n,displaystyle I_3:=-frac 12int _0^1int _0^1frac P_n(x)P_n(y)log(xy)1-xydx,dy=b_nzeta (3)-a_n,

where |I|≤ζ(3)(1−2)4ndisplaystyle displaystyle , Pn(z)displaystyle P_n(z)P_n(z) are the Legendre polynomials, and the subsequences bn,2lcm⁡(1,2,…,n)⋅an∈Zdisplaystyle b_n,2operatorname lcm (1,2,ldots ,n)cdot a_nin mathbb Z displaystyle b_n,2operatorname lcm (1,2,ldots ,n)cdot a_nin mathbb Z are integers or almost integers.


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



Series representations



Classical


In 1772, Leonhard Euler gave the series representation:[7]


ζ(3)=π27(1−4∑k=1∞ζ(2k)22k(2k+1)(2k+2))displaystyle zeta (3)=frac pi ^27left(1-4sum _k=1^infty frac zeta (2k)2^2k(2k+1)(2k+2)right)displaystyle zeta (3)=frac pi ^27left(1-4sum _k=1^infty frac zeta (2k)2^2k(2k+1)(2k+2)right)

which was subsequently rediscovered several times.[8]


Other classical series representations include:


ζ(3)=87∑k=0∞1(2k+1)3ζ(3)=43∑k=0∞(−1)k(k+1)3displaystyle beginalignedzeta (3)&=frac 87sum _k=0^infty frac 1(2k+1)^3\zeta (3)&=frac 43sum _k=0^infty frac (-1)^k(k+1)^3endaligneddisplaystyle beginalignedzeta (3)&=frac 87sum _k=0^infty frac 1(2k+1)^3\zeta (3)&=frac 43sum _k=0^infty frac (-1)^k(k+1)^3endaligned


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,[9] then rediscovered and widely advertised by Apéry in 1979:[3]


ζ(3)=52∑k=1∞(−1)k−1k!2(2k)!k3=52∑k=1∞(−1)k−1(2kk)k3displaystyle beginalignedzeta (3)&=frac 52sum _k=1^infty (-1)^k-1frac k!^2(2k)!k^3\&=frac 52sum _k=1^infty frac (-1)^k-1binom 2kkk^3endaligneddisplaystyle beginalignedzeta (3)&=frac 52sum _k=1^infty (-1)^k-1frac k!^2(2k)!k^3\&=frac 52sum _k=1^infty frac (-1)^k-1binom 2kkk^3endaligned

The following series representation, found by Amdeberhan in 1996,[10] gives (asymptotically) 1.43 new correct decimal places per term:


ζ(3)=14∑k=1∞(−1)k−156k2−32k+5(2k−1)2(k−1)!3(3k)!displaystyle zeta (3)=frac 14sum _k=1^infty (-1)^k-1frac 56k^2-32k+5(2k-1)^2frac (k-1)!^3(3k)!displaystyle zeta (3)=frac 14sum _k=1^infty (-1)^k-1frac 56k^2-32k+5(2k-1)^2frac (k-1)!^3(3k)!

The following series representation, found by Amdeberhan and Zeilberger in 1997,[11] gives (asymptotically) 3.01 new correct decimal places per term:


ζ(3)=∑k=0∞(−1)k205k2+250k+7764k!10(2k+1)!5displaystyle zeta (3)=sum _k=0^infty (-1)^kfrac 205k^2+250k+7764frac k!^10(2k+1)!^5zeta(3) = sum_k=0^infty (-1)^k frac205k^2 + 250k + 7764 frack!^10(2k+1)!^5

The following series representation, found by Sebastian Wedeniwski in 1998,[12] gives (asymptotically) 5.04 new correct decimal places per term:


ζ(3)=∑k=0∞(−1)k((2k+1)!(2k)!k!)324(3k+2)!(4k+3)!3P(k)displaystyle zeta (3)=sum _k=0^infty (-1)^kfrac big ((2k+1)!(2k)!k!big )^324(3k+2)!(4k+3)!^3,P(k)displaystyle zeta (3)=sum _k=0^infty (-1)^kfrac big ((2k+1)!(2k)!k!big )^324(3k+2)!(4k+3)!^3,P(k)

where


P(k)=126392k5+412708k4+531578k3+336367k2+104000k+12463.displaystyle P(k)=126,392k^5+412,708k^4+531,578k^3+336,367k^2+104,000k+12,463.displaystyle 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.[13]


The following series representation, found by Mohamud Mohammed in 2005,[14] gives (asymptotically) 3.92 new correct decimal places per term:


ζ(3)=12∑k=0∞(−1)k(2k)!3(k+1)!6(k+1)2(3k+3)!4P(k)displaystyle zeta (3)=frac 12,sum _k=0^infty frac (-1)^k(2k)!^3(k+1)!^6(k+1)^2(3k+3)!^4,P(k)displaystyle zeta (3)=frac 12,sum _k=0^infty frac (-1)^k(2k)!^3(k+1)!^6(k+1)^2(3k+3)!^4,P(k)

where


P(k)=40885k5+124346k4+150160k3+89888k2+26629k+3116.displaystyle P(k)=40,885k^5+124,346k^4+150,160k^3+89,888k^2+26,629k+3116.,displaystyle P(k)=40,885k^5+124,346k^4+150,160k^3+89,888k^2+26,629k+3116.,


Digit by digit


In 1998, Broadhurst[15] 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:[16]


ζ(3)=7180π3−2∑k=1∞1k3(e2πk−1)displaystyle zeta (3)=frac 7180pi ^3-2sum _k=1^infty frac 1k^3(e^2pi k-1)displaystyle zeta (3)=frac 7180pi ^3-2sum _k=1^infty frac 1k^3(e^2pi k-1)

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


ζ(3)=14∑k=1∞1k3sinh⁡(πk)−112∑k=1∞1k3(e2πk−1)−72∑k=1∞1k3(e2πk+1).displaystyle zeta (3)=14sum _k=1^infty frac 1k^3sinh(pi k)-frac 112sum _k=1^infty frac 1k^3(e^2pi k-1)-frac 72sum _k=1^infty frac 1k^3(e^2pi k+1).displaystyle zeta (3)=14sum _k=1^infty frac 1k^3sinh(pi k)-frac 112sum _k=1^infty frac 1k^3(e^2pi k-1)-frac 72sum _k=1^infty frac 1k^3(e^2pi k+1).

Srivastava[18] 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:



ζ(3)=∫01∫01∫0111−xyzdxdydzdisplaystyle zeta (3)=int _0^1!!int _0^1!!int _0^1frac 11-xyz,dx,dy,dzdisplaystyle zeta (3)=int _0^1!!int _0^1!!int _0^1frac 11-xyz,dx,dy,dz.

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


ζ(3)=12∫0∞x2ex−1dxdisplaystyle zeta (3)=frac 12int _0^infty frac x^2e^x-1,dxdisplaystyle zeta (3)=frac 12int _0^infty frac x^2e^x-1,dx

and



ζ(3)=23∫0∞x2ex+1dxdisplaystyle zeta (3)=frac 23int _0^infty frac x^2e^x+1,dxdisplaystyle zeta (3)=frac 23int _0^infty frac x^2e^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:


ζ(3)=47∫0π2xlog⁡(sec⁡x+tan⁡x)dxdisplaystyle zeta (3)=frac 47int _0^frac pi 2xlog (sec x+tan x),dxdisplaystyle zeta (3)=frac 47int _0^frac pi 2xlog (sec x+tan x),dx

Note the similarity to


G=12∫0π2log⁡(sec⁡x+tan⁡x)dxdisplaystyle G=frac 12int _0^frac pi 2log (sec x+tan x),dxdisplaystyle G=frac 12int _0^frac pi 2log (sec x+tan x),dx

where G is Catalan's constant.



More complicated formulas


For example, one formula was found by Johan Jensen:[19]



ζ(3)=π∫0∞cos⁡(2arctan⁡x)(x2+1)(cosh⁡12πx)2dxdisplaystyle zeta (3)=pi !!int _0^infty !frac cos(2arctan x)left(x^2+1right)left(cosh frac 12pi xright)^2,dxdisplaystyle zeta (3)=pi !!int _0^infty !frac cos(2arctan x)left(x^2+1right)left(cosh frac 12pi xright)^2,dx,

another by F. Beukers:[5]



ζ(3)=−12∫01∫01ln⁡(xy)1−xydxdy=−∫01∫01ln⁡(1−xy)xydxdydisplaystyle zeta (3)=-frac 12int _0^1!!int _0^1frac ln(xy),1-xy,,dx,dy=-int _0^1!!int _0^1frac ln(1-xy),xy,,dx,dydisplaystyle zeta (3)=-frac 12int _0^1!!int _0^1frac ln(xy),1-xy,,dx,dy=-int _0^1!!int _0^1frac ln(1-xy),xy,,dx,dy,

Mixing these two formula, one can obtain :
ζ(3)=∫01ln⁡(x)ln⁡(1−x)xdxdisplaystyle zeta (3)=int _0^1!!frac ln(x)ln(1-x),x,,dxdisplaystyle zeta (3)=int _0^1!!frac ln(x)ln(1-x),x,,dx


and yet another by Iaroslav Blagouchine:[20]



ζ(3)=8π27∫01x(x4−4x2+1)ln⁡ln⁡1x(1+x2)4dx=8π27∫1∞x(x4−4x2+1)ln⁡ln⁡x(1+x2)4dxdisplaystyle beginalignedzeta (3)&=frac 8pi ^27!!int _0^1!frac xleft(x^4-4x^2+1right)ln ln frac 1x,(1+x^2)^4,,dx\&=frac 8pi ^27!!int _1^infty !frac xleft(x^4-4x^2+1right)ln ln x,(1+x^2)^4,,dxendaligneddisplaystyle beginalignedzeta (3)&=frac 8pi ^27!!int _0^1!frac xleft(x^4-4x^2+1right)ln ln frac 1x,(1+x^2)^4,,dx\&=frac 8pi ^27!!int _1^infty !frac xleft(x^4-4x^2+1right)ln ln x,(1+x^2)^4,,dxendaligned.

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


ζ(3)=−12Γ‴(1)+32Γ′(1)Γ″(1)−(Γ′(1))3=−12ψ(2)(1)displaystyle zeta (3)=-tfrac 12Gamma '''(1)+tfrac 32Gamma '(1)Gamma ''(1)-big (Gamma '(1)big )^3=-tfrac 12,psi ^(2)(1)displaystyle zeta (3)=-tfrac 12Gamma '''(1)+tfrac 32Gamma '(1)Gamma ''(1)-big (Gamma '(1)big )^3=-tfrac 12,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)
DateDecimal digitsComputation performed by
173516
Leonhard Euler
unknown16
Adrien-Marie Legendre
188732
Thomas Joannes Stieltjes
19967005520000000000000♠520000Greg J. Fee & Simon Plouffe
19977006100000000000000♠1000000Bruno Haible & Thomas Papanikolaou
May 19977007105360060000000♠10536006Patrick Demichel
February 19987007140000740000000♠14000074Sebastian Wedeniwski
March 19987007320002130000000♠32000213Sebastian Wedeniwski
July 19987007640000910000000♠64000091Sebastian Wedeniwski
December 19987008128000026000000♠128000026Sebastian Wedeniwski[1]
September 20017008200001000000000♠200001000Shigeru Kondo & Xavier Gourdon
February 20027008600001000000000♠600001000Shigeru Kondo & Xavier Gourdon
February 20037009100000000000000♠1000000000Patrick Demichel & Xavier Gourdon[22]
April 20067010100000000000000♠10000000000Shigeru Kondo & Steve Pagliarulo
January 20097010155100000000000♠15510000000Alexander J. Yee & Raymond Chan[23]
March 20097010310260000000000♠31026000000Alexander J. Yee & Raymond Chan[23]
September 20107011100000001000000♠100000001000Alexander J. Yee[24]
September 20137011200000001000000♠200000001000Robert J. Setti[24]
August 20157011250000000000000♠250000000000Ron Watkins[24]
November 20157011400000000000000♠400000000000Dipanjan Nag[25]


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).[26]



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.[27] In 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[28]



See also


  • Riemann zeta function


  • Basel problem — ζ(2)

  • List of sums of reciprocals


Notes




  1. ^ ab See Wedeniwski 2001.


  2. ^ See Frieze 1985.


  3. ^ ab See Apéry 1979.


  4. ^ See van der Poorten 1979.


  5. ^ ab See Beukers 1979.


  6. ^ See Zudilin 2002.


  7. ^ See Euler 1773.


  8. ^ See Srivastava 2000, p. 571 (1.11).


  9. ^ See Hjortnaes 1953.


  10. ^ See Amdeberhan 1996.


  11. ^ See Amdeberhan & Zeilberger 1997.


  12. ^ 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).


  13. ^ See Wedeniwski 1998 and Wedeniwski 2001.


  14. ^ See Mohammed 2005.


  15. ^ See Broadhurst 1998.


  16. ^ See Berndt 1989, chapter 14, formulas 25.1 and 25.3.


  17. ^ See Plouffe 1998.


  18. ^ See Srivastava 2000.


  19. ^ See Jensen 1895.


  20. ^ See Blagouchine 2014.


  21. ^ See Evgrafov et al. 1969, exercise 30.10.1.


  22. ^ See Gourdon & Sebah 2003.


  23. ^ ab See Yee 2009.


  24. ^ abc See Yee 2015.


  25. ^ See Nag 2015.


  26. ^ Mollin (2009).


  27. ^ See Rivoal 2000.


  28. ^ See Zudilin 2001.



References


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  • Yee, Alexander J. (2015), Zeta(3) - Apéry's Constant


  • Nag, Dipanjan (2015), Calculated Apéry's constant to 400,000,000,000 Digit, A world record


  • Zudilin, Wadim (2001), "One of the numbers ζ(5)displaystyle zeta (5)zeta (5), ζ(7)displaystyle zeta (7)displaystyle zeta (7), ζ(9)displaystyle zeta (9)displaystyle zeta (9), ζ(11)displaystyle zeta (11)displaystyle zeta (11) is irrational", Russ. Math. Surv., 56 (4): 774–776, Bibcode:2001RuMaS..56..774Z, doi:10.1070/RM2001v056n04ABEH000427.


  • Zudilin, Wadim (2002), An elementary proof of Apéry's theorem, arXiv:math/0202159, Bibcode:2002math......2159Z.



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










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