Apéry's constant

In mathematics, Apéry's constant is the infinite sum of the reciprocals of the positive integers, cubed. That is, it is defined as the number

$\begin{array}{rl}\zeta (3)& =\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}\\ & =\underset{n\to \mathrm{\infty }}{\lim }(\frac{1}{1^3}+\frac{1}{2^3}+\cdots +\frac{1}{n^3}),\end{array}$

where Template:Mvar is the Riemann zeta function. It has an approximate value ofTemplate:Sfnp

ζ(3) ≈ Template:Gaps (sequence A002117 in the OEIS).

It is named after Roger Apéry, who proved that it is an irrational number.

Uses

Apéry's 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 treesTemplate:Sfnp 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.

The reciprocal of ζ(3) (0.8319073725807... (sequence A088453 in the OEIS)) is the probability that any three positive integers, chosen at random, will be relatively prime, in the sense that as N approaches infinity, the probability that three positive integers less than N chosen uniformly at random will not share a common prime factor approaches this value. (The probability for n positive integers is 1/ζ(n).Template:Sfnp) In the same sense, it is the probability that a positive integer chosen at random will not be evenly divisible by the cube of an integer greater than one. (The probability for not having divisibility by an n-th power is 1/ζ(n).Template:Sfnp)

Properties

Unsolved problem in mathematics

Is Apéry's constant transcendental?

More unsolved problems in mathematics

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

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3),

$\zeta (3)=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{1-xyz}dxdydz,$

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

$I_3:=-\frac{1}{2}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{P_n(x)P_n(y)\log (xy)}{1-xy}dxdy=b_n\zeta (3)-a_n,$

where $|I|\le \zeta (3)(1-\sqrt{2}{)}^{4n}$, $P_n(z)$ are the Legendre polynomials, and the subsequences $b_n,2\mathrm{lcm}(1,2,\dots ,n)\cdot a_n\in \mathbb{\mathbb{Z}}$ are integers or almost integers.

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational.Template:Sfnp In particular at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.Template:Sfnp

Apéry's constant has not yet been proved transcendental, but it is known to be an algebraic period. This follows immediately from the form of its triple integral.

Series representations

Classical

In addition to the fundamental series:

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

Leonhard Euler gave the series representation:Template:Sfnp

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

in 1772, which was subsequently rediscovered several times.Template:Sfnp

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,Template:Sfnp rediscovered by Hjortnaes in 1953,Template:Sfnp and rediscovered once more and widely advertised by Apéry in 1979:Template:Sfnp

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

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:Template:Sfnp

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

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:Template:Sfnp

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

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

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

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

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:Template:Sfnp

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

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 by a spigot algorithm in nearly linear time and logarithmic space.Template:Sfnp

Thue-Morse sequence

Apéry's constant can be represented in terms of the Thue-Morse sequence $(t_n{)}_{n\ge 0}$, as follows:

$\begin{array}{rl}\sum _{n\ge 1}\frac{9t_{n-1}+7t_n}{n^3}& =8\zeta (3),\end{array}$

This is a special case of the following formula (valid for all $s$ with real part greater than $1$):

$(2^s+1)\sum _{n\ge 1}\frac{t_{n-1}}{n^s}+(2^s-1)\sum _{n\ge 1}\frac{t_n}{n^s}=2^s\zeta (s).$

Others

The following series representation was found by Ramanujan:

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

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

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

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

The following formula follows directly from the integral definition of the zeta function:

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

More complicated formulas

Other formulas includeTemplate:Sfnp

$\zeta (3)=\pi \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (2\arctan x)}{(x^2+1){(\cosh \frac{1}{2}\pi x)}^2}dx$

andTemplate:Sfnp

$\zeta (3)=-\frac{1}{2}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{\log (xy)}{1-xy}dxdy=-\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{\log (1-xy)}{xy}dxdy.$

Also,Template:Sfnp

$\begin{array}{rl}\zeta (3)& =\frac{8{\pi }^2}{7}\underset{0}{\overset{1}{\int }}\frac{x(x^4-4x^2+1)\log \log \frac{1}{x}}{(1+x^2{)}^4}dx\\ & =\frac{8{\pi }^2}{7}\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{x(x^4-4x^2+1)\log \log x}{(1+x^2{)}^4}dx.\end{array}$

A connection to the derivatives of the gamma function

$\zeta (3)=-\frac{1}{2}({\mathrm{\Gamma }}^{‴}(1)+{\gamma }^3+\frac{1}{2}{\pi }^2\gamma )=-\frac{1}{2}{\psi }^{(2)}(1)$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.Template:Sfnp

Continued fraction

Apéry's constant is related to the following continued fraction:

$\frac{6}{\zeta (3)}=5-\frac{1}{117-\frac{64}{535-\frac{729}{1436-\frac{4096}{3105-\frac{15625}{\dots }}}}}$

with $a_n=34n^3+51n^2+27n+5$ and $b_n=-n^6$.

Its simple continued fraction is given by:

$\zeta (3)=1+\frac{1}{4+\frac{1}{1+\frac{1}{18+\frac{1}{1+\frac{1}{1+\frac{1}{\dots }}}}}}$

Known digits

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades, and now stands at more than . 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)

Date	Decimal digits	Computation performed by
1735	16	Leonhard Euler
Unknown	16	Adrien-Marie Legendre
1887	32	Thomas Joannes Stieltjes
1996		Greg J. Fee & Simon Plouffe
1997		Bruno Haible & Thomas Papanikolaou
May 1997		Patrick Demichel
February 1998		Sebastian Wedeniwski
March 1998		Sebastian Wedeniwski
July 1998		Sebastian Wedeniwski
December 1998		Sebastian WedeniwskiTemplate:Sfnp
September 2001		Shigeru Kondo & Xavier Gourdon
February 2002		Shigeru Kondo & Xavier Gourdon
February 2003		Patrick Demichel & Xavier GourdonTemplate:Sfnp
April 2006		Shigeru Kondo & Steve Pagliarulo
January 21, 2009		Alexander J. Yee & Raymond ChanTemplate:Sfnp
February 15, 2009		Alexander J. Yee & Raymond ChanTemplate:Sfnp
September 17, 2010		Alexander J. YeeTemplate:Sfnp
September 23, 2013		Robert J. SettiTemplate:Sfnp
August 7, 2015		Ron WatkinsTemplate:Sfnp
December 21, 2015		Dipanjan NagTemplate:Sfnp
August 13, 2017		Ron WatkinsTemplate:Sfnp
May 26, 2019		Ian Cutress
July 26, 2020		Seungmin Kim
December 22, 2023		Andrew Sun

See also

• Riemann zeta function
• Basel problem — ζ(2)
• Catalan's constant
• List of sums of reciprocals

References

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• (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).
• (Message to Simon Plouffe, with original text but only some decimal places).
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Further reading

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External links

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