Apéry's Constant

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Apéry's constant is defined by

zeta(3)=1.2020569...,
(1)

(OEIS A002117) where zeta(z) is the Riemann zeta function. Apéry (1979) proved that zeta(3) is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of zeta(3) (Hata 2000). zeta(3) arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.

The following table summarizes progress in computing upper bounds on the irrationality measure for zeta(3). Here, the exact values for mu_4 is given by

mu_4=1+(4ln(sqrt(2)+1)+3)/(4ln(sqrt(2)+1)-3)
(2)
approx 13.4178202
(3)

(Hata 2000).

mu_nupper boundreference
15.513891Rhin and Viola (2001)
28.830284Hata (1990)
312.74359Dvornicich and Viola (1987)
413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)

Beukers (1979) reproduced Apéry's rational approximation to zeta(3) using the triple integral of the form

int_0^1int_0^1int_0^1(L_n(x)L_n(y))/(1-(1-xy)u)dxdydu,
(4)

where L_n(x) is a Legendre polynomial. Beukers's integral is given by

zeta(3)=-1/2int_0^1int_0^1(ln(xy))/((1-xy))dxdy,
(5)

a result that is a special case of what is known as Hadjicostas's formula.

This integral is closely related to zeta(3) using the curious identity

int_0^1int_0^1int_0^1(x^ry^s)/(1-(1-xy)u)dxdydu={2zeta(3)-sum_(l=1)^(r)2/(l^3) for r=s; sum_(l=min(r,s)+1)^(max(r,s))1/(|r-s|l^2) for r!=s
(6)
={2zeta(3)-2H_r^((3)) for r=s; (psi_1(1+min(r,s))-psi_1(1+max(r,s)))/(|r-s|) for r!=s,
(7)

where H_r^((n)) is a generalized harmonic number and psi_k(x) is a polygamma function (Hata 2000).

Sums related to zeta(3) include

zeta(3)=5/2sum_(n=1)^(infty)((-1)^(n-1))/(n^3(2n; n))
(8)
=5/2sum_(k=1)^(infty)((-1)^(k+1)(k!)^2)/((2k)!k^3)
(9)

(used by Apéry), the related sum

zeta(3)=2/3(ln2)^3+4sum_(k=1)^infty((-1)^(k+1))/(k^32^k(2k; k))
(10)

as first proved by G. Huvent in 2002 (Gourevitch) and rediscovered by B. Cloitre (pers. comm., Oct. 8, 2004), and

sum_(k=0)^(infty)1/((2k+1)^3)=7/8zeta(3)
(11)
=lambda(3)
(12)
sum_(k=0)^(infty)1/((3k+1)^3)=(2pi^3)/(81sqrt(3))+(13)/(27)zeta(3)
(13)
sum_(k=0)^(infty)1/((4k+1)^3)=(pi^3)/(64)+7/(16)zeta(3)
(14)
sum_(k=0)^(infty)1/((6k+1)^3)=(pi^3)/(36sqrt(3))+(91)/(216)zeta(3),
(15)

where lambda(z) is the Dirichlet lambda function. The above equations are special cases of a general result due to Ramanujan (Berndt 1985).

Apéry's constant is given by an infinite family BBP-type formulas of the form

zeta(3)=4/3sum_(k=0)^(infty)((-1)^k)/((1+k)^3)
(16)
=4/3sum_(k=0)^(infty)(-1)^k[1/((3k+1)^3)-1/((3k+2)^3)+1/((3k+3)^3)]
(17)
=3/2sum_(k=0)^(infty)(-1)^k[1/((3k+1)^3)-1/((3k+2)^3)-2/((3k+3)^3)]
(18)
=4/3sum_(k=0)^(infty)(-1)^k[1/((5k+1)^3)-1/((5k+2)^3)+1/((5k+3)^3)-1/((5k+4)^3)+1/((5k+5)^3)]
(19)
=1/(15)sum_(k=0)^(infty)(-1)^k[(21)/((5k+1)^3)-(21)/((5k+2)^3)+(21)/((5k+3)^3)-(21)/((5k+4)^3)-(104)/((5k+5)^3)]
(20)
=4/3sum_(k=0)^(infty)(-1)^k[1/((7k+1)^3)-1/((7k+2)^3)+1/((7k+3)^3)-1/((7k+4)^3)+1/((7k+5)^3)-1/((7k+6)^3)+1/((7k+7)^3)]
(21)
=1/(30)sum_(k=0)^(infty)(-1)^k[(41)/((7k+1)^3)-(41)/((7k+2)^3)+(41)/((7k+3)^3)-(41)/((7k+4)^3)+(41)/((7k+5)^3)-(41)/((7k+6)^3)+(302)/((7k+7)^3)]
(22)

(E. W. Weisstein, Feb. 25, 2006), and the amazing two special sums

zeta(3)=1/(672)sum_(k=0)^(infty)1/(4096^k)[(2048)/((24k+1)^3)-(11264)/((24k+2)^3)-(1024)/((24k+3)^3)+(11776)/((24k+4)^3)-(512)/((24k+5)^3)+(4096)/((24k+6)^3)+(256)/((24k+7)^3)+(3456)/((24k+8)^3)+(128)/((24k+9)^3)-(704)/((24k+10)^3)-(64)/((24k+11)^3)-(128)/((24k+12)^3)-(32)/((24k+13)^3)-(176)/((24k+14)^3)+(16)/((24k+15)^3)+(216)/((24k+16)^3)+8/((24k+17)^3)+(64)/((24k+18)^3)-4/((24k+19)^3)+(46)/((24k+20)^3)-2/((24k+21)^3)-(11)/((24k+22)^3)+1/((24k+23)^3)]
(23)
=9/(224)sum_(k=0)^(infty)1/(4096^k)[(1024)/((24k+2)^3)-(3072)/((24k+3)^3)+(512)/((24k+4)^3)+(1024)/((24k+6)^3)+(1152)/((24k+8)^3)+(384)/((24k+9)^3)+(64)/((24k+10)^3)+(128)/((24k+12)^3)+(16)/((24k+14)^3)+(48)/((24k+15)^3)+(72)/((24k+16)^3)+(16)/((24k+18)^3)+2/((24k+20)^3)-6/((24k+21)^3)+1/((24k+22)^3)].
(24)

Determining a sum of this type is given as an exercise by Bailey et al. (2007, p. 225; typo corrected).

A beautiful double series for zeta(3) is given by

zeta(3)=1/3sum_(i=1)^inftysum_(j=1)^infty((i-1)!(j-1)!)/((i+j)!)H_(i+j),
(25)

where H_n is a harmonic number (O. Oloa, pers. comm., Dec. 30, 2005).

Apéry's proof relied on showing that the sum

a(n)=sum_(k=0)^n(n; k)^2(n+k; k)^2,
(26)

where (n; k) is a binomial coefficient, satisfies the recurrence relation

n^3a_n-(34n^3-51n^2+27n-5)a_(n-1)+(n-1)^3a_(n-2)=0
(27)

(van der Poorten 1979, Zeilberger 1991). The characteristic polynomial x^2-34x+1 has roots (1+/-sqrt(2))^4, so

lim_(n->infty)(a_(n+1))/(a_n)=(1+sqrt(2))^4
(28)

is irrational and a_n cannot satisfy a two-term recurrence (Jin and Dickinson 2000).

Apéry's constant is also given by

zeta(3)=8sum_(n=1)^infty(S_(n,2))/(n!n),
(29)

where S_(n,m) is a Stirling number of the first kind. This can be rewritten as

zeta(3)=1/2sum_(n=1)^(infty)1/(n^2)(1+1/2+...+1/n)
(30)
=1/2sum_(n=1)^(infty)(H_n)/(n^2),
(31)

where H_n is the nth harmonic number (Castellanos 1988).

Integrals for zeta(3) include

zeta(3)=1/2int_0^infty(t^2)/(e^t-1)dt
(32)
=8/7[1/4pi^2ln2+2int_0^(pi/2)xln(sinx)dx].
(33)

Gosper (1990) gave

zeta(3)=1/4sum_(k=1)^infty(30k-11)/((2k-1)k^3(2k; k)^2).
(34)

A continued fraction involving Apéry's constant is

6/(zeta(3))=5-(1^6)/(117-)(2^6)/(535-)...(n^6)/(34n^3+51n^2+27n+5-)...
(35)

(Apéry 1979, Le Lionnais 1983). Amdeberhan (1996) used Wilf-Zeilberger pairs (F,G) with

F(n,k)=((-1)^kk!^2(sn-k-1)!)/((sn+k+1)!(k+1)),
(36)

s=1 to obtain

zeta(3)=5/2sum_(n=1)^infty(-1)^(n-1)1/((2n; n)n^3).
(37)

For s=2,

zeta(3)=1/4sum_(n=1)^infty(-1)^(n-1)(56n^2-32n+5)/((2n-1)^2)1/((3n; n)(2n; n)n^3)
(38)

(Boros and Moll 2004, p. 236; Amdeberhan 1996), and for s=3,

zeta(3)=sum_(n=0)^infty((-1)^n)/(72(4n; n)(3n; n))(5265n^4+13878n^3+13761n^2+6120n+1040)/((4n+1)(4n+3)(n+1)(3n+1)^2(3n+2)^2)
(39)

(Amdeberhan 1996). The corresponding G(n,k) for s=1 and 2 are

G(n,k)=(2(-1)^kk!^2(n-k)!)/((n+k+1)!(n+1)^2)
(40)

and

G(n,k)=((-1)^kk!^2(2n-k)!(3+4n)(4n^2+6n+k+3))/(2(2n+k+2)!(n+1)^2(2n+1)^2).
(41)

zeta(3) is related to the Glaisher-Kinkelin constant A and polygamma function psi_n(z) by

zeta(3)=2/3pi^2[12psi_(-4)(1)-6lnA-ln(2pi)].
(42)

Gosper (1996) expressed zeta(3) as the matrix product

lim_(N->infty)product_(n=1)^NM_n=[0 zeta(3); 0 1],
(43)

where

M_n=[((n+1)^4)/(4096(n+5/4)^2(n+7/4)^2) (24570n^4+64161n^3+62152n^2+26427n+4154)/(31104(n+1/3)(n+1/2)(n+2/3)); 0 1]
(44)

which gives 12 bits per term. The first few terms are

M_1=[1/(19600) (2077)/(1728); 0 1]
(45)
M_2=[1/(9801) (7561)/(4320); 0 1]
(46)
M_3=[9/(67600) (50501)/(20160); 0 1],
(47)

which gives

zeta(3) approx (423203577229)/(352066176000)=1.20205690315732....
(48)

Given three integers chosen at random, the probability that no common factor will divide them all is

[zeta(3)]^(-1) approx 1.20206^(-1) approx 0.831907.
(49)

B. Haible and T. Papanikolaou computed zeta(3) to 1000000 digits using a Wilf-Zeilberger pair identity with

F(n,k)=(-1)^k(n!^6(2n-k-1)!k!^3)/(2(n+k+1)!^2(2n)!^3),
(50)

s=1, and t=1, giving the rapidly converging

zeta(3)=sum_(n=0)^infty(-1)^n(n!^(10)(205n^2+250n+77))/(64(2n+1)!^5)
(51)

(Amdeberhan and Zeilberger 1997). The record as of Dec. 1998 was 128 million digits, computed by S. Wedeniwski.

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