Apéry's constant

Template:Irrational numbers
Binary	Template:Gaps
Decimal	Template:Gaps
Hexadecimal	Template:Gaps
Continued fraction	$1+{\frac {1}{4+{\cfrac {1}{1+{\cfrac {1}{18+{\cfrac {1}{\ddots \qquad {}}}}}}}}}$ Note that this continued fraction is infinite, but it is not known whether this continued fraction is periodic or not.

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

${\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}$

where Template:Math is the Riemann zeta function. It has an approximate value of

Template:Math Template: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 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

Template:Math 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 $\zeta (3)$,

$\zeta (3)=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz,$

by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that

$I_{3}:=-{\frac {1}{2}}\int _{0}^{1}\int _{0}^{1}{\frac {P_{n}(x)P_{n}(y)\log(xy)}{1-xy}}\,dx\,dy=b_{n}\zeta (3)-a_{n},$

where $|I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}$, $P_{n}(z)$ are the Legendre polynomials, and the subsequences $b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z}$ 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:

$\zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},$

Leonhard Euler gave the series representation:

$\zeta (3)={\frac {\pi ^{2}}{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)$

in 1772, which was subsequently rediscovered several times.

Other classical series representations include:

${\begin{aligned}\zeta (3)&={\frac {8}{7}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)^{3}}}\\\zeta (3)&={\frac {4}{3}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(k+1)^{3}}}\end{aligned}}$

Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of Template:Math. 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:

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

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

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

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

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

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

$\zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}$

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:

$\zeta (3)={\frac {1}{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}$

where

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:

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

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

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

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:

$\zeta (3)=\int _{0}^{1}\!\!\int _{0}^{1}\!\!\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz$.

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

$\zeta (3)={\frac {1}{2}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx$

and

$\zeta (3)={\frac {2}{3}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}+1}}\,dx$.

This one follows from a Taylor expansion of Template:Math about Template:Math, where Template:Math is the Legendre chi function:

$\zeta (3)={\frac {4}{7}}\int _{0}^{\frac {\pi }{2}}x\log {(\sec {x}+\tan {x})}\,dx$

Note the similarity to

$G={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\log {(\sec {x}+\tan {x})}\,dx$

where Template:Mvar is Catalan's constant.

More complicated formulas

For example, one formula was found by Johan Jensen:

$\zeta (3)=\pi \!\!\int _{0}^{\infty }\!{\frac {\cos(2\arctan {x})}{\left(x^{2}+1\right)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx$,

another by F. Beukers:

$\zeta (3)=-{\frac {1}{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\ln(xy)}{\,1-xy\,}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\ln(1-xy)}{\,xy\,}}\,dx\,dy$,

Mixing these two formula, one can obtain : $\zeta (3)=\int _{0}^{1}\!\!{\frac {\ln(x)\ln(1-x)}{\,x\,}}\,dx$

and yet another by Iaroslav Blagouchine:

${\begin{aligned}\zeta (3)&={\frac {8\pi ^{2}}{7}}\!\!\int _{0}^{1}\!{\frac {x\left(x^{4}-4x^{2}+1\right)\ln \ln {\frac {1}{x}}}{\,(1+x^{2})^{4}\,}}\,dx\\&={\frac {8\pi ^{2}}{7}}\!\!\int _{1}^{\infty }\!{\frac {x\left(x^{4}-4x^{2}+1\right)\ln \ln {x}}{\,(1+x^{2})^{4}\,}}\,dx\end{aligned}}$.

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

$\zeta (3)=-{\tfrac {1}{2}}\Gamma '''(1)+{\tfrac {3}{2}}\Gamma '(1)\Gamma ''(1)-{\big (}\Gamma '(1){\big )}^{3}=-{\tfrac {1}{2}}\,\psi ^{(2)}(1)$

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 Template:Math 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 Template:Math

Date	Decimal digits	Computation performed by
1735	16	Leonhard Euler
unknown	16	Adrien-Marie Legendre
1887	32	Thomas Joannes Stieltjes
1996	Template:Val	Greg J. Fee & Simon Plouffe
1997	Template:Val	Bruno Haible & Thomas Papanikolaou
May 1997	Template:Val	Patrick Demichel
February 1998	Template:Val	Sebastian Wedeniwski
March 1998	Template:Val	Sebastian Wedeniwski
July 1998	Template:Val	Sebastian Wedeniwski
December 1998	Template:Val	Sebastian Wedeniwski
September 2001	Template:Val	Shigeru Kondo & Xavier Gourdon
February 2002	Template:Val	Shigeru Kondo & Xavier Gourdon
February 2003	Template:Val	Patrick Demichel & Xavier Gourdon
April 2006	Template:Val	Shigeru Kondo & Steve Pagliarulo
January 2009	Template:Val	Alexander J. Yee & Raymond Chan
March 2009	Template:Val	Alexander J. Yee & Raymond Chan
September 2010	Template:Val	Alexander J. Yee
September 2013	Template:Val	Robert J. Setti
August 2015	Template:Val	Ron Watkins
November 2015	Template:Val	Dipanjan Nag
August 2017	Template:Val	Ron Watkins
June 2019	Template:Val	Ian Curtess

Reciprocal

The reciprocal of Template:Math is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as Template:Math goes to infinity, the probability that three positive integers less than Template:Math chosen uniformly at random will be relatively prime approaches this value).Template:Sfnp

Extension to Template:Math

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

See also

• Riemann zeta function
• Basel problem — Template:Math
• List of sums of reciprocals

References

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