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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). Apéry's proof involved use of the continued fraction

6/(zeta(3))=5+K_(n=1)^infty(-n^6)/(17[n^3+(n+1)^3]-12(2n+1))
(2)

(Raayoni 2021, Elimelech et al. 2023).

Lai (2026) reported that the numbers 1, zeta(3), and zeta(4) are linearly independent over the rationals. Since zeta(4)=pi^4/90, this implies that zeta(3)/pi^4 is irrational. Lai (2026) states that ChatGPT GPT-6 Astra generated the proof and that he verified and rewrote it, taking responsibility for its correctness.

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.

Leibtag et al. (2026) obtained the improved irrationality measure bound

mu(zeta(3))<5.4197.
(3)

This improves the earlier bound 5.513891 of Rhin and Viola (2001). They also gave an independent proof that 1, zeta(3), and zeta(4) are linearly independent, with mu(Azeta(3)+Bzeta(4))<66.3 for every nonzero pair of rational numbers (A,B). Leibtag et al. (2026) state that their main independence construction was developed without an LLM. They report later AI assistance with technical arithmetic estimates used for the improved bounds and with formalization.

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)
(4)
approx 13.4178202
(5)

(Hata 2000).

mu_nupper boundreference
15.513891Rhin and Viola (2001)
15.4197Leibtag et al. (2026)
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,
(6)

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,
(7)

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
(8)
={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,
(9)

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))
(10)
=5/2sum_(k=1)^(infty)((-1)^(k+1)(k!)^2)/((2k)!k^3)
(11)

(used by Apéry), the related sum

zeta(3)=2/3(ln2)^3+4sum_(k=1)^infty((-1)^(k+1))/(k^32^k(2k; k))
(12)

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)
(13)
=lambda(3)
(14)
sum_(k=0)^(infty)1/((3k+1)^3)=(2pi^3)/(81sqrt(3))+(13)/(27)zeta(3)
(15)
sum_(k=0)^(infty)1/((4k+1)^3)=(pi^3)/(64)+7/(16)zeta(3)
(16)
sum_(k=0)^(infty)1/((6k+1)^3)=(pi^3)/(36sqrt(3))+(91)/(216)zeta(3),
(17)

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)
(18)
=4/3sum_(k=0)^(infty)(-1)^k[1/((3k+1)^3)-1/((3k+2)^3)+1/((3k+3)^3)]
(19)
=3/2sum_(k=0)^(infty)(-1)^k[1/((3k+1)^3)-1/((3k+2)^3)-2/((3k+3)^3)]
(20)
=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)]
(21)
=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)]
(22)
=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)]
(23)
=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)]
(24)

(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)]
(25)
=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)].
(26)

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

Rapidly converging series include

zeta(3)=7/(180)pi^3-2sum_(k=1)^infty1/(k^3(e^(2pik)-1))
(27)

(Berndt 1988, Plouffe 1998) and

zeta(3)=(pi^3)/(28)+(16)/7sum_(n=1)^infty1/(n^3(e^(npi)+1))-2/7sum_(n=1)^infty1/(n^3(e^(2pin)+1))
(28)

(Plouffe 2006).

A sum limit is

zeta(3)=lim_(x->infty)1/((2x+1)^3)sum_(k=1)^x[cot(k/(2x+1))]^3
(29)

(Apostol 1973, given incorrectly in Stark 1974), where x tends to infinity through positive integers.

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),
(30)

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,
(31)

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
(32)

(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
(33)

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

Apéry's constant is also given by

zeta(3)=8sum_(n=1)^infty(S_(n,2))/(n!n),
(34)

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)
(35)
=1/2sum_(n=1)^(infty)(H_n)/(n^2),
(36)

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

Amdeberhan (1996) used Wilf-Zeilberger pairs (F,G) with

F(n,k)=((-1)^kk!^2(sn-k-1)!)/((sn+k+1)!(k+1)),
(37)

s=1 to obtain

zeta(3)=5/2sum_(n=1)^infty(-1)^(n-1)1/((2n; n)n^3).
(38)

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)
(39)

(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)
(40)

(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)
(41)

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).
(42)

Amdeberhan and Zeilberger (1997) used 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),
(43)

s=1, and t=1, to obtain the rapidly converging series

zeta(3)=sum_(n=0)^infty(-1)^n((n!)^(10)(205n^2+250n+77))/(64((2n+1)!)^5),
(44)

which was used to compute zeta(3) to 1 million decimal digits. Campbell (2022) used the WZ method to obtain

zeta(3)=-2/7-1/(448)sum_(n=1)^infty((-2^(12))^n(7168n^5-1664n^4-1328n^3+212n^2+49n-9))/(n^4(2n-1)(3n+1)(4n+1)(2n; n)(3n; n)(4n; 2n)^3).
(45)

Zuniga (2023, p. 7) gives the rapidly converging series

zeta(3)=1/(48)sum_(n=1)^infty((-1)^(n-1)P(n))/(R(n)(5n; n)(5n; 2n)(9n; 4n)(10n; 5n)(12n; 6n)),
(46)

where

P(n)=1565994397644288n^(11)-6719460725627136n^(10)+12632254526031264n^9-13684352515879536n^8+9451223531851808n^7-4348596587040104n^6+1352700034136826n^5-282805786014979n^4+38721705264979n^3-3292502315430n^2+156286859400n-3143448000,
(47)

and

R(n)=n^5(2n-1)^3(3n-1)(3n-2)(4n-1)(4n-3)(6n-1)(6n-5).
(48)

The ratio of successive terms tends to -1/717445350000, corresponding to about 11.86 decimal digits per term.

Integrals for zeta(3) include

zeta(3)=1/2int_0^infty(t^2)/(e^t-1)dt
(49)
=8/7[1/4pi^2ln2+2int_0^(pi/2)xln(sinx)dx].
(50)

Gosper (1990) gave

zeta(3)=1/4sum_(k=1)^infty(30k-11)/((2k-1)k^3(2k; k)^2).
(51)

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-)...
(52)

(Apéry 1979, Le Lionnais 1983).

The derivative of the Riemann zeta function satisfies

zeta^'(-2)=-(zeta(3))/(4pi^2).
(53)

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)].
(54)

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

lim_(N->infty)product_(n=1)^NM_n=[0 zeta(3); 0 1],
(55)

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]
(56)

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

M_1=[1/(19600) (2077)/(1728); 0 1]
(57)
M_2=[1/(9801) (7561)/(4320); 0 1]
(58)
M_3=[9/(67600) (50501)/(20160); 0 1],
(59)

which gives

zeta(3) approx (423203577229)/(352066176000)=1.20205690315732....
(60)

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.
(61)

See also

Apéry's Constant Approximations, Apéry's Constant Continued Fraction, Apéry's Constant Digits, Apéry-Constant Prime, Hadjicostas's Formula, Planck's Radiation Function, Riemann Zeta Function, Riemann Zeta Function zeta(2), Trilogarithm, Wilf-Zeilberger Pair

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References

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Apéry's Constant

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Weisstein, Eric W. "Apéry's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AperysConstant.html

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