Apéry's constant
| Binary | 1.0011001110111010… |
| Decimal | 1.2020569031595942854… |
| Hexadecimal | 1.33BA004F00621383… |
| Continued fraction |
Note that this continued fraction is infinite, but it is not known whether this continued fraction is periodic or not. |
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
This 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 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.
It is still not known whether Apéry's constant is transcendental.
In 1772, Leonhard Euler gave the series representation:
which was subsequently rediscovered several times.
Other classical series representations include:
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").
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