Apéry's constant

In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number where ζ is the Riemann zeta function. It has an approximate value of 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 for the French mathematician Roger Apéry, who proved in 1978 that it is irrational. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later. Beuker's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ ( 3 ) {displaystyle zeta (3)} , by the Legendre polynomials.In particular, van der Poorten's article chronicles this approach by noting that where | I | ≤ ζ ( 3 ) ( 1 − 2 ) 4 n {displaystyle |I|leq zeta (3)(1-{sqrt {2}})^{4n}} , P n ( z ) {displaystyle P_{n}(z)} are the Legendre polynomials, and the subsequences b n , 2 lcm ⁡ ( 1 , 2 , … , n ) ⋅ a n ∈ Z {displaystyle b_{n},2operatorname {lcm} (1,2,ldots ,n)cdot a_{n}in mathbb {Z} } are integers or almost integers. It is still not known whether Apéry's constant is transcendental.