Apéry's constant - Alchetron, The Free Social Encyclopedia

In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number

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.

Irrational number

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.

Classical

In 1772, Leonhard Euler gave the series representation:

ζ ( 3 ) = π 2 7 ( 1 − 4 ∑ k = 1 ∞ ζ ( 2 k ) 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) )

which was subsequently rediscovered several times.

Other classical series representations include:

ζ ( 3 ) = 8 7 ∑ k = 0 ∞ 1 ( 2 k + 1 ) 3 ζ ( 3 ) = 4 3 ∑ k = 0 ∞ ( − 1 ) k ( k + 1 ) 3

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 Hjortnaes in 1953, then rediscovered and widely advertised by Apéry in 1979:

ζ ( 3 ) = 5 2 ∑ k = 1 ∞ ( − 1 ) k − 1 k ! 2 ( 2 k ) ! k 3 = 5 2 ∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 k k ) k 3

The following series representation, found by Amdeberhan in 1996, gives (asymptotically) 1.43 new correct decimal places per term:

ζ ( 3 ) = 1 4 ∑ k = 1 ∞ ( − 1 ) k − 1 56 k 2 − 32 k + 5 ( 2 k − 1 ) 2 ( k − 1 ) ! 3 ( 3 k ) !

The following series representation, found by Amdeberhan and Zeilberger in 1997, gives (asymptotically) 3.01 new correct decimal places per term:

ζ ( 3 ) = ∑ k = 0 ∞ ( − 1 ) k 205 k 2 + 250 k + 77 64 k ! 10 ( 2 k + 1 ) ! 5

The following series representation, found by Sebastian Wedeniwski in 1998, gives (asymptotically) 5.04 new correct decimal places per term:

ζ ( 3 ) = ∑ k = 0 ∞ ( − 1 ) k ( ( 2 k + 1 ) ! ( 2 k ) ! k ! ) 3 24 ( 3 k + 2 ) ! ( 4 k + 3 ) ! 3 P ( k )

where

P ( k ) = 126 392 k 5 + 412 708 k 4 + 531 578 k 3 + 336 367 k 2 + 104 000 k + 12 463.

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 (asymptotically) 3.92 new correct decimal places per term:

ζ ( 3 ) = 1 2 ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! 3 ( k + 1 ) ! 6 ( k + 1 ) 2 ( 3 k + 3 ) ! 4 P ( k )

where

P ( k ) = 40 885 k 5 + 124 346 k 4 + 150 160 k 3 + 89 888 k 2 + 26 629 k + 3116.

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:

ζ ( 3 ) = 7 180 π 3 − 2 ∑ k = 1 ∞ 1 k 3 ( e 2 π k − 1 )

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

ζ ( 3 ) = 14 ∑ k = 1 ∞ 1 k 3 sinh ⁡ ( π k ) − 11 2 ∑ k = 1 ∞ 1 k 3 ( e 2 π k − 1 ) − 7 2 ∑ k = 1 ∞ 1 k 3 ( e 2 π k + 1 ) .

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:

ζ ( 3 ) = ∫ 0 1 ∫ 0 1 ∫ 0 1 1 1 − x y z d x d y d z .

The next two follow directly from the well-known integral formulas for the Riemann zeta function:

ζ ( 3 ) = 1 2 ∫ 0 ∞ x 2 e x − 1 d x

and

ζ ( 3 ) = 2 3 ∫ 0 ∞ x 2 e x + 1 d x .

This one follows from a Taylor expansion of χ₃(e^ix) about x = ±π/2, where χ_ν(z) is the Legendre chi function:

ζ ( 3 ) = 4 7 ∫ 0 π 2 x log ⁡ ( sec ⁡ x + tan ⁡ x ) d x

Note the similarity to

G = 1 2 ∫ 0 π 2 log ⁡ ( sec ⁡ x + tan ⁡ x ) d x

where G is Catalan's constant.

More complicated formulas

For example, one formula was found by Johan Jensen:

ζ ( 3 ) = π ∫ 0 ∞ cos ⁡ ( 2 arctan ⁡ x ) ( x 2 + 1 ) ( cosh ⁡ 1 2 π x ) 2 d x ,

another by F. Beukers:

ζ ( 3 ) = − 1 2 ∫ 0 1 ∫ 0 1 ln ⁡ ( x y ) 1 − x y d x d y ,

and yet another by Iaroslav Blagouchine:

ζ ( 3 ) = 8 π 2 7 ∫ 0 1 x ( x 4 − 4 x 2 + 1 ) ln ⁡ ln ⁡ 1 x ( 1 + x 2 ) 4 d x = 8 π 2 7 ∫ 1 ∞ x ( x 4 − 4 x 2 + 1 ) ln ⁡ ln ⁡ x ( 1 + x 2 ) 4 d x .

Evgrafov et al.'s connection to the derivatives of the gamma function

ζ ( 3 ) = − 1 2 Γ ‴ ( 1 ) + 3 2 Γ ′ ( 1 ) Γ ″ ( 1 ) − ( Γ ′ ( 1 ) ) 3 = − 1 2 ψ ( 2 ) ( 1 )

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 ζ(3) has increased dramatically during the last decades. This is due both to the increasing performance of computers and to algorithmic improvements.

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).

Extension to ζ(2n + 1)

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other odd zeta values. In 2000, Tanguy Rivoal showed that infinitely many of the numbers ζ(2n + 1) must be irrational. In 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.

Credits

This article incorporates material from Apéry's constant on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

References

Apéry's constant Wikipedia

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