Apéry's constant

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number where ζ is the Riemann zeta function. It has an approximate value of ζ(3) = 1.20205 69031 59594 28539 97381 61511 44999 07649 86292 ... . The constant is named after Roger Apéry. It 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 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. ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later. 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 , are the Legendre polynomials, and the subsequences are integers or almost integers. It is still not known whether Apéry's constant is transcendental. In addition to the fundamental series: Leonhard Euler gave the series representation: in 1772, which was subsequently rediscovered several times. 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"). The following series representation was found by A. A. Markov in 1890, rediscovered by Hjortnaes in 1953, and rediscovered once more and widely advertised by Apéry in 1979: The following series representation gives (asymptotically) 1.