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.
DateDecimal digitsComputation performed by
173516Leonhard Euler
unknown16Adrien-Marie Legendre
188732Thomas Joannes Stieltjes
1996Greg J. Fee & Simon Plouffe
1997Bruno Haible & Thomas Papanikolaou
May 1997Patrick Demichel
February 1998Sebastian Wedeniwski
March 1998Sebastian Wedeniwski
July 1998Sebastian Wedeniwski
December 1998Sebastian Wedeniwski
September 2001Shigeru Kondo & Xavier Gourdon
February 2002Shigeru Kondo & Xavier Gourdon
February 2003Patrick Demichel & Xavier Gourdon
April 2006Shigeru Kondo & Steve Pagliarulo
January 2009Alexander J. Yee & Raymond Chan
March 2009Alexander J. Yee & Raymond Chan
September 2010Alexander J. Yee
September 2013Robert J. Setti
August 2015Ron Watkins
November 2015Dipanjan Nag
August 2017Ron Watkins
June 2019Ian Cutress

Reciprocal

The reciprocal of is the probability that any three positive integers, chosen at random, will be relatively prime.

Extension to

Many people have tried to extend Apéry's proof that is irrational to other values of the zeta function with odd arguments. In 2000, Tanguy Rivoal showed that infinitely many of the numbers must be irrational. In 2001, Wadim Zudilin proved that at least one of the numbers,,, and must be irrational.

Credits