Apéry's constant

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

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

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

ζ(3) ≈ 1.202056903159594285399738161511449990764986292… (sequence A002117 in the OEIS).

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

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 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.

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).^[3]) 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).^[3])

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.^[4] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,^[5] and simpler proofs were found later.^[6]

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

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

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

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

where ${\displaystyle |I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}}$, ${\displaystyle P_{n}(z)}$ are the Legendre polynomials, and the subsequences ${\displaystyle b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z} }$ 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.^[7] In particular at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.^[8]

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.

In addition to the fundamental series:

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

Leonhard Euler gave the series representation:^[9]

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

in 1772, which was subsequently rediscovered several times.^[10]

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,^[11] rediscovered by Hjortnaes in 1953,^[12] and rediscovered once more and widely advertised by Apéry in 1979:^[4]

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

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:^[13]

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

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:^[14]

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

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:^[15]

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

It has been used to calculate Apéry's constant with several million correct decimal places.^[16]

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:^[17]

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

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.^[18]

The following representation was found by Tóth in 2022:^[19]

${\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}$

where ${\displaystyle (t_{n})_{n\geq 0}}$ is the ${\displaystyle n^{\rm {th}}}$ term of the Thue-Morse sequence. In fact, this is a special case of the following formula (valid for all ${\displaystyle s}$ with real part greater than ${\displaystyle 1}$):

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

The following series representation was found by Ramanujan:^[20]

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

The following series representation was found by Simon Plouffe in 1998:^[21]

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

Srivastava (2000) collected many series that converge to Apéry's constant.

There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

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

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

Other formulas include^[22]

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

and^[23]

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

Also,^[24]

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

A connection to the derivatives of the gamma function^[25]

${\displaystyle \zeta (3)=-{\tfrac {1}{2}}(\Gamma '''(1)+\gamma ^{3}+{\tfrac {1}{2}}\pi ^{2}\gamma )=-{\tfrac {1}{2}}\psi ^{(2)}(1)}$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.^[26]

Apéry's constant is related to the following continued fraction:^[27]

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

with ${\displaystyle a_{n}=34n^{3}+51n^{2}+27n+5}$ and ${\displaystyle b_{n}=-n^{6}}$.

Its simple continued fraction is given by:^[28]

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

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades, and now stands at more than 2×10¹². 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	520000	Greg J. Fee & Simon Plouffe
1997	1000000	Bruno Haible & Thomas Papanikolaou
May 1997	10536006	Patrick Demichel
February 1998	14000074	Sebastian Wedeniwski
March 1998	32000213	Sebastian Wedeniwski
July 1998	64000091	Sebastian Wedeniwski
December 1998	128000026	Sebastian Wedeniwski[1]
September 2001	200001000	Shigeru Kondo & Xavier Gourdon
February 2002	600001000	Shigeru Kondo & Xavier Gourdon
February 2003	1000000000	Patrick Demichel & Xavier Gourdon[29]
April 2006	10000000000	Shigeru Kondo & Steve Pagliarulo
January 21, 2009	15510000000	Alexander J. Yee & Raymond Chan[30]
February 15, 2009	31026000000	Alexander J. Yee & Raymond Chan[30]
September 17, 2010	100000001000	Alexander J. Yee[31]
September 23, 2013	200000001000	Robert J. Setti[31]
August 7, 2015	250000000000	Ron Watkins[31]
December 21, 2015	400000000000	Dipanjan Nag[32]
August 13, 2017	500000000000	Ron Watkins[31]
May 26, 2019	1000000000000	Ian Cutress[33]
July 26, 2020	1200000000100	Seungmin Kim[33][34]
December 22, 2023	2020569031595	Andrew Sun[33]

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

• Ramaswami, V. (1934), "Notes on Riemann's ${\displaystyle \zeta }$-function", J. London Math. Soc., 9 (3): 165–169, doi:10.1112/jlms/s1-9.3.165.
• Nahin, Paul J. (2021), In pursuit of zeta-3 : the world's most mysterious unsolved math problem, Princeton: Princeton University Press, ISBN 978-0-691-22759-7, OCLC 1260168397

• Weisstein, Eric W., "Apéry's constant", MathWorld{{cite web}}: CS1 maint: overridden setting (link)
• Plouffe, Simon, Zeta(3) or Apéry constant to 2000 places, archived from the original on 2008-02-05, retrieved 2005-07-29
• Setti, Robert J. (2015), Apéry's Constant - Zeta(3) - 200 Billion Digits, archived from the original on 2013-10-08.

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