In mathematics, Apéry's constant is the infinite sum of the reciprocals of the positive integers, cubed.
That is, it is defined as the number where ζ is the Riemann zeta function.
It has an approximate value of[1] It is named after Roger Apéry, who proved that it is an irrational number.
Apéry's 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.
The reciprocal of ζ(3) (0.8319073725807... (sequence A088453 in the OEIS)) is the probability that any three positive integers, chosen at random, will be relatively prime, in the sense that as N approaches infinity, the probability that three positive integers less than N chosen uniformly at random will not share a common prime factor approaches this value.
[6] Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3), by the Legendre polynomials.
In particular, van der Poorten's article chronicles this approach by noting that where
Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments.
Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational.
[8] Apéry's constant has not yet been proved transcendental, but it is known to be an algebraic period.
[10] 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").
A. Markov in 1890,[11] rediscovered by Hjortnaes in 1953,[12] and rediscovered once more and widely advertised by Apéry in 1979:[4] The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[13] The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[14] The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[15] It has been used to calculate Apéry's constant with several million correct decimal places.
[16] The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[17] In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained by a spigot algorithm in nearly linear time and logarithmic space.
Its simple continued fraction is given by:[28] The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades, and now stands at more than 2×1012.
This is due both to the increasing performance of computers and to algorithmic improvements.