Apéry's Constant

Apéry's constant is defined by

(1)

(OEIS A002117) where  is the Riemann zeta function. Apéry (1979) proved that  is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of  (Hata 2000).  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 . Here, the exact values for  is given by

			(2)
			(3)

(Hata 2000).

	upper bound	reference
1	5.513891	Rhin and Viola (2001)
2	8.830284	Hata (1990)
3	12.74359	Dvornicich and Viola (1987)
4	13.41782	Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)

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

(4)

where  is a Legendre polynomial. Beukers's integral is given by

(5)

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

This integral is closely related to  using the curious identity

			(6)
			(7)

where  is a generalized harmonic number and  is a polygamma function (Hata 2000).

Sums related to  include

			(8)
			(9)

(used by Apéry), the related sum

(10)

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

			(11)
			(12)
			(13)
			(14)
			(15)

where  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

			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

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

			(23)
			(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  is given by

(25)

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

Apéry's proof relied on showing that the sum

(26)

where  is a binomial coefficient, satisfies the recurrence relation

(27)

(van der Poorten 1979, Zeilberger 1991). The characteristic polynomial  has roots , so

(28)

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

Apéry's constant is also given by

(29)

where  is a Stirling number of the first kind. This can be rewritten as

			(30)
			(31)

where  is the th harmonic number (Castellanos 1988).

Integrals for  include

			(32)
			(33)

Gosper (1990) gave

(34)

A continued fraction involving Apéry's constant is

(35)

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

(36)

 to obtain

(37)

For ,

(38)

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

(39)

(Amdeberhan 1996). The corresponding  for  and 2 are

(40)

and

(41)

 is related to the Glaisher-Kinkelin constant  and polygamma function  by

(42)

Gosper (1996) expressed  as the matrix product

(43)

where

(44)

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

			(45)
			(46)
			(47)

which gives

(48)

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

(49)

B. Haible and T. Papanikolaou computed  to  digits using a Wilf-Zeilberger pair identity with

(50)

, and , giving the rapidly converging

(51)

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

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