Apéry's constant

In mathematics, at the intersection of number theory and special functions, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

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.

where ζ is the Riemann zeta function. It has an approximate value of[1]

ζ(3) = 1.202056903159594285399738161511449990764986292 (sequence A002117 in the OEIS).

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[2] 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

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.[3] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[4] and simpler proofs were found later.[5]

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.

Series representations

Classical

In addition to the fundamental series:

Leonhard Euler gave the series representation:[6]

in 1772, which was subsequently rediscovered several times.[7]

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 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,[8] rediscovered by Hjortnaes in 1953,[9] and rediscovered once more and widely advertised by Apéry in 1979:[3]

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[10]

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[11]

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[12]

It has been used to calculate Apéry's constant with several million correct decimal places.[13]

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[14]

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.[15]

Others

The following series representation was found by Ramanujan:[16]

The following series representation was found by Simon Plouffe in 1998:[17]

Srivastava (2000) 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 χ3(eix) about x = ±π/2, where χν(z) is the Legendre chi function:

Note the similarity to

where G is Catalan's constant.

More complicated formulas

Other formulas include:[18]

,

and,[19]

,

Mixing these two formulas, one can obtain :

,

By symmetry,

,

Summing both, .

Also,[20]

.

A 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.[21]

Known digits

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increasing performance of computers and to algorithmic improvements.

Number of known decimal digits of Apéry's constant ζ(3)
DateDecimal digitsComputation performed by
173516Leonhard Euler
unknown16Adrien-Marie Legendre
188732Thomas Joannes Stieltjes
1996520000Greg J. Fee & Simon Plouffe
19971000000Bruno Haible & Thomas Papanikolaou
May 199710536006Patrick Demichel
February 199814000074Sebastian Wedeniwski
March 199832000213Sebastian Wedeniwski
July 199864000091Sebastian Wedeniwski
December 1998128000026Sebastian Wedeniwski[1]
September 2001200001000Shigeru Kondo & Xavier Gourdon
February 2002600001000Shigeru Kondo & Xavier Gourdon
February 20031000000000Patrick Demichel & Xavier Gourdon[22]
April 200610000000000Shigeru Kondo & Steve Pagliarulo
January 21, 200915510000000Alexander J. Yee & Raymond Chan[23]
February 15, 200931026000000Alexander J. Yee & Raymond Chan[23]
September 17, 2010100000001000Alexander J. Yee[24]
September 23, 2013200000001000Robert J. Setti[24]
August 7, 2015250000000000Ron Watkins[24]
December 21, 2015400000000000Dipanjan Nag[25]
August 13, 2017500000000000Ron Watkins[24]
May 26, 20191000000000000Ian Cutress[26]
July 26, 20201200000000100Seungmin Kim[27][28]

Reciprocal

The reciprocal of ζ(3) is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as N goes to infinity, the probability that three positive integers less than N chosen uniformly at random will be relatively prime approaches this value).[29]

Extension to ζ(2n + 1)

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the zeta function with odd arguments. Infinitely many of the numbers ζ(2n + 1) must be irrational,[30] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[31]

See also

Notes

References

Further reading

  • Ramaswami, V. (1934), "Notes on Riemann's -function", J. London Math. Soc., 9 (3): 165–169, doi:10.1112/jlms/s1-9.3.165.

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