Apéry's constant is defined by
(1)
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(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). Apéry's proof involved use of the continued fraction
(2)
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(Raayoni 2021, Elimelech et al. 2023).
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
(3)
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(4)
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(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
(5)
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where is a Legendre polynomial. Beukers's integral is given by
(6)
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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
(7)
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(8)
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where is a generalized harmonic number and is a polygamma function (Hata 2000).
Sums related to include
(9)
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(10)
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(used by Apéry), the related sum
(11)
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as first proved by G. Huvent in 2002 (Gourevitch) and rediscovered by B. Cloitre (pers. comm., Oct. 8, 2004), and
(12)
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(13)
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(14)
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(15)
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(16)
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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
(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(E. W. Weisstein, Feb. 25, 2006), and the amazing two special sums
(24)
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(25)
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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
(26)
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where is a harmonic number (O. Oloa, pers. comm., Dec. 30, 2005).
Apéry's proof relied on showing that the sum
(27)
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where is a binomial coefficient, satisfies the recurrence relation
(28)
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(van der Poorten 1979, Zeilberger 1991). The characteristic polynomial has roots , so
(29)
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is irrational and cannot satisfy a two-term recurrence (Jin and Dickinson 2000).
Apéry's constant is also given by
(30)
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where is a Stirling number of the first kind. This can be rewritten as
(31)
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(32)
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where is the th harmonic number (Castellanos 1988).
Amdeberhan (1996) used Wilf-Zeilberger pairs with
(33)
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to obtain
(34)
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For ,
(35)
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(Boros and Moll 2004, p. 236; Amdeberhan 1996), and for ,
(36)
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(Amdeberhan 1996). The corresponding for and 2 are
(37)
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and
(38)
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Amdeberhan and Zeilberger (1997) used a Wilf-Zeilberger pair identity with
(39)
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, and , to obtain the rapidly converging series
(40)
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which was used to compute to 1 million decimal digits. Campbell (2022) used the WZ method to obtain
(41)
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Integrals for include
(42)
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(43)
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Gosper (1990) gave
(44)
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A continued fraction involving Apéry's constant is
(45)
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(Apéry 1979, Le Lionnais 1983).
is related to the Glaisher-Kinkelin constant and polygamma function by
(46)
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Gosper (1996) expressed as the matrix product
(47)
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where
(48)
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which gives 12 bits per term. The first few terms are
(49)
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(50)
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(51)
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which gives
(52)
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Given three integers chosen at random, the probability that no common factor will divide them all is
(53)
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