Apéry's constant

In mathematics, Apéry's constant is a curious number that occurs in a variety of situations. It rises 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 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. It is defined as the number $zeta(3)$,

:$zeta(3)=sum\_\{k=1\}^inftyfrac\{1\}\{k^3\}=1+frac\{1\}\{2^3\}\; +\; frac\{1\}\{3^3\}\; +frac\{1\}\{4^3\}\; +\; cdots$

where ζ is the Riemann zeta function. It has an approximate value of harv|Wedeniwski|2001

:$zeta(3)=1.20205;\; 69031;\; 59594;\; 28539;\; 97381;61511;\; 44999;\; 07649;\; 86292,ldots$ OEIS|id=A002117

The reciprocal of this number 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 < N chosen uniformly at random will be relatively prime approaches this value).

Apéry's theorem

This value was named for Roger Apéry (1916–1994), who in 1978 proved it to be irrational. This result is known as "Apéry's theorem". The original proof is complex and hard to grasp, and shorter proofs have been found later, using Legendre polynomials. It is not known whether Apéry's constant is transcendental.

Work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2"n"+1) must be irrational, [T. Rivoal, "La fonction zeta de Riemann prend une infnité de valuers irrationnelles aux entiers impairs", Comptes Rendus Acad. Sci. Paris Sér. I Math. 331 (2000), pp. 267-270.] and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational. [W. Zudilin, "One of the numbers ζ(5); ζ(7); ζ(9); ζ(11) is irrational", Uspekhi Mat. Nauk 56:4 (2001), pp. 149-150.]

eries representation

In 1772, Leonhard Euler harv|Euler|1773 gave the series representation harv|Srivastava|2000|loc=p. 571 (1.11):

:$zeta(3)=frac\{pi^2\}\{7\}left\; [\; 1-4sum\_\{k=1\}^infty\; frac\; \{zeta\; (2k)\}\; \{(2k+1)(2k+2)\; 2^\{2k\; ight]$

which was subsequently rediscovered several times.

Simon Plouffe gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include harv|Plouffe|1998:

:$zeta(3)=frac\{7\}\{180\}pi^3\; -2\; sum\_\{k=1\}^infty\; frac\{1\}\{k^3\; (e^\{2pi\; k\}\; -1)\}$

and

:$zeta(3)=\; 14\; sum\_\{k=1\}^infty\; frac\{1\}\{k^3\; sinh(pi\; k)\}-frac\{11\}\{2\}sum\_\{k=1\}^infty\; frac\{1\}\{k^3\; (e^\{2pi\; k\}\; -1)\}-frac\{7\}\{2\}\; sum\_\{k=1\}^infty\; frac\{1\}\{k^3\; (e^\{2pi\; k\}\; +1)\}.$

Similar relations for the values of $zeta(2n+1)$ are given in the article zeta constants.

Many additional series representations have been found, including:

:$zeta(3)\; =\; frac\{8\}\{7\}\; sum\_\{k=0\}^infty\; frac\{1\}\{(2k+1)^3\}$

:$zeta(3)\; =\; frac\{4\}\{3\}\; sum\_\{k=0\}^infty\; frac\{(-1)^k\}\{(k+1)^3\}$

:$zeta(3)\; =\; frac\{5\}\{2\}\; sum\_\{k=1\}^infty\; (-1)^\{k-1\}\; frac\{(k!)^2\}\{k^3\; (2k)!\}$

:$zeta(3)\; =\; frac\{1\}\{4\}\; sum\_\{k=1\}^infty\; (-1)^\{k-1\}frac\{56k^2-32k+5\}\{(2k-1)^2\}\; frac\{((k-1)!)^3\}\{(3k)!\}$

:$zeta(3)=frac\{8\}\{7\}-frac\{8\}\{7\}sum\_\{k=1\}^infty\; frac$left( -1 ight) }^k,2^{-5 + 12,k},k, left( -3 + 9,k + 148,k^2 - 432,k^3 - 2688,k^4 + 7168,k^5 ight) , {k!}^3,{left( -1 + 2,k ight) !}^6}left( -1 + 2,k ight) }^3, left( 3,k ight) !,{left( 1 + 4,k ight) !}^3}

:$zeta(3)\; =\; sum\_\{k=0\}^infty\; (-1)^k\; frac\{205nk^2\; +\; 250k\; +\; 77\}\{64\}\; frac\{(k!)^\{10${((2k+1)!)^5}

and

:$zeta(3)\; =\; sum\_\{k=0\}^infty\; (-1)^k\; frac\{P(k)\}\{24\}frac\{((2k+1)!(2k)!k!)^3\}\{(3k+2)!((4k+3)!)^3\}$

where

:$P(k)\; =\; 126392k^5\; +\; 412708k^4\; +\; 531578k^3\; +\; 336367k^2\; +\; 104000k\; +\; 12463.,$

Some of these have been used to calculate Apéry's constant with several million digits.

harvtxt|Broadhurst|1998 gives 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.

Other formulas

Apéry's constant can be expressed in terms of the second-order polygamma function as

:$zeta(3)\; =\; -frac\{1\}\{2\}\; ,\; psi^\{(2)\}(1).$

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 increase of performance of computers as well as to algorithmic improvements.

References

*citation|first=D.J.|last=Broadhurst|url=http://arxiv.org/abs/math.CA/9803067|title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)|year=1998|publisher=arXiv (math.CA/9803067).
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* citation
last = Srivastava
first = H. M.
year = 2000
month = December
title = Some Families of Rapidly Convergent Series Representations for the Zeta Functions
journal = Taiwanese Journal of Mathematics
volume = 4
issue = 4
pages = 569–598
publisher = Mathematical Society of the Republic of China (Taiwan)
issn = 1027-5487
oclc =36978119
url = http://www.math.nthu.edu.tw/~tjm/abstract/0012/tjm0012_3.pdf
format = PDF
accessdate = 2008-05-18
*citation
last = Euler
first = Leonhard
authorlink = Leonhard Euler
year = 1773
title = Exercitationes analyticae
journal = Novi Commentarii academiae scientiarum Petropolitanae
volume = 17
pages = 173–204
url = http://math.dartmouth.edu/~euler/docs/originals/E432.pdf
language = Latin
format = PDF
accessdate = 2008-05-18
*
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