Apéry's constant
In mathematics, at the intersection of number theory and special functions, Apéry's constant is the sum of the inverses of the positive cubes. That is, it is defined as the number
where 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.
Irrational number
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.Beuker's simplified irrationality proof involves approximating the integrand of the known triple integral for,
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.
Series representations
Classical
In addition to the fundamental series:Leonhard Euler gave the series representation:
in 1772, which was subsequently rediscovered several times.
Other classical series representations include:
Fast convergence
Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of. Since the 1990s, this search has focused on computationally efficient series with fast convergence rates.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, found by Amdeberhan in 1996, gives 1.43 new correct decimal places per term:
The following series representation, found by Amdeberhan and Zeilberger in 1997, gives 3.01 new correct decimal places per term:
The following series representation, found by Sebastian Wedeniwski in 1998, gives 5.04 new correct decimal places per term:
It was used by Wedeniwski to calculate Apéry's constant with several million correct decimal places.
The following series representation, found by Mohamud Mohammed in 2005, gives 3.92 new correct decimal places per term:
Digit by digit
In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.Others
The following series representation was found by Ramanujan:The following series representation was found by Simon Plouffe in 1998:
Srivastava collected many series that converge to Apéry's constant.
Integral representations
There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.Simple formulas
For example, this one follows from the summation representation for Apéry's constant:The next two follow directly from the well-known integral formulas for the Riemann zeta function:
and
This one follows from a Taylor expansion of about, where is the Legendre chi function:
Note the similarity to
where is Catalan's constant.
More complicated formulas
For example, one formula was found by Johan Jensen:another by F. Beukers:
Mixing these two formulas, one can obtain :
By symmetry,
Summing both,
Yet another by Iaroslav Blagouchine:
Evgrafov et al.'s connection to the derivatives of the gamma function
is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma-functions.
Known digits
The number of known digits of Apéry's constant has increased dramatically during the last decades. This is due both to the increasing performance of computers and to algorithmic improvements.| Date | Decimal digits | Computation performed by |
| 1735 | 16 | Leonhard Euler |
| unknown | 16 | Adrien-Marie Legendre |
| 1887 | 32 | Thomas Joannes Stieltjes |
| 1996 | Greg J. Fee & Simon Plouffe | |
| 1997 | Bruno Haible & Thomas Papanikolaou | |
| May 1997 | Patrick Demichel | |
| February 1998 | Sebastian Wedeniwski | |
| March 1998 | Sebastian Wedeniwski | |
| July 1998 | Sebastian Wedeniwski | |
| December 1998 | Sebastian Wedeniwski | |
| September 2001 | Shigeru Kondo & Xavier Gourdon | |
| February 2002 | Shigeru Kondo & Xavier Gourdon | |
| February 2003 | Patrick Demichel & Xavier Gourdon | |
| April 2006 | Shigeru Kondo & Steve Pagliarulo | |
| January 2009 | Alexander J. Yee & Raymond Chan | |
| March 2009 | Alexander J. Yee & Raymond Chan | |
| September 2010 | Alexander J. Yee | |
| September 2013 | Robert J. Setti | |
| August 2015 | Ron Watkins | |
| November 2015 | Dipanjan Nag | |
| August 2017 | Ron Watkins | |
| June 2019 | Ian Cutress |