Apéry's constant

Apéry's constant

Rationality	Irrational
Symbol	ζ(3)
Representations
Decimal	1.2020569031595942854...
Continued fraction	$1+{\frac {1}{4+{\cfrac {1}{1+{\cfrac {1}{18+{\cfrac {1}{\ddots \qquad {}}}}}}}}}$ Unknown whether periodic Infinite
Binary	1.0011001110111010...
Hexadecimal	1.33BA004F00621383...

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it 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 ζ is the Riemann zeta function. It has an approximate value of

ζ(3) = 1.202056903159594285399738161511449990764986292… (sequence A002117 in the 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

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. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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.

Thue-Morse sequence

The following representation was found by Tóth in 2022:

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

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

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

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 (2000) 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.

More complicated formulas

Other formulas include

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

and

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

Also,

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

A connection to the derivatives of the gamma function

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

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 ζ(3) 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 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
September 2001	200001000	Shigeru Kondo & Xavier Gourdon
February 2002	600001000	Shigeru Kondo & Xavier Gourdon
February 2003	1000000000	Patrick Demichel & Xavier Gourdon
April 2006	10000000000	Shigeru Kondo & Steve Pagliarulo
January 21, 2009	15510000000	Alexander J. Yee & Raymond Chan
February 15, 2009	31026000000	Alexander J. Yee & Raymond Chan
September 17, 2010	100000001000	Alexander J. Yee
September 23, 2013	200000001000	Robert J. Setti
August 7, 2015	250000000000	Ron Watkins
December 21, 2015	400000000000	Dipanjan Nag
August 13, 2017	500000000000	Ron Watkins
May 26, 2019	1000000000000	Ian Cutress
July 26, 2020	1200000000100	Seungmin Kim

Reciprocal

The reciprocal of ζ(3) (0.8319073725807... (sequence A088453 in the OEIS)) comes up in the context of probabilistic number theory: This is due to its expansion as a product indexed by the primes, together with the fact that a natural number $n<N$ drawn uniformly has probability $1-1/p+o_{N}(1)$ of not being divisible by a given prime p, and the fact that, due to the chinese remainder theorem, the events of not being divisible by finitely many different primes are independent, save a constant of size o(1). As a trivial example designed to arrive at Apéry's constant, note that the probability that a natural number less than N drawn uniformly will not be divisible by the cube of an integer greater than one is $\prod _{p<N^{1/6}}(1-1/p^{3})+o(1)$. Since this converges to $\zeta (3)^{-1}$ we get heuristics that indicates that the natural density of the natural numbers that are not divisible by any cube equals the reciprocal of Apéry's constant. Here, the reason that we have to talk about natural density rather than probability for this subset of the naturals is that the uniform probability measure on the subset the naturals n<N as defined above will not become a probability measure on the naturals when we let N tend to $\infty$ (to see this, consider what happens if we assign a point probability zero, and a non-zero probability respectively, and remember that a probability measure has to be countable additive). One therefore speaks of natural density instead. Moreover, the reason that it still does not constitute a proof, but instead a useful heuristics, is that we would need a version of the chinese remainder theorem that holds for infinitely many primes and that extends the analogue of the product formula from probability theory to hold also for natural densities.

Summing up: We have a uniform probability measure on the subsets of the natural numbers less N for any N. But letting N tend to infinity we do not get a probability measure on the natural numbers as it will not be countably additive (the subsets does not form a sigma-algebra). Instead we get the natural density. And we cannot use tools from probability theory when computing natural densities. Nevertheless, we can apply tools from probability to conjecture formulas, which can then be proven by other means. Here we gave a simple example where this heuristic is utilized to conjectures that the natural density of the natural numbers that are not divisible by any cube is equal to the reciprocal of Apéry's constant. A more substantial example when this heuristic was used is the Erdős–Kac theorem (which would not be possible to conjecture by computing values of the function in question, see that article). This is also known as the fundamental theorem of probabilistic number theory; it was conjectured by Mark Kac using the above heuristics and proved by Paul Erdős using sieve theory. See Erdős–Kac theorem for more details and references.

Extension to ζ(2n + 1)

Main article: Particular values of the Riemann zeta function

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the zeta function with odd arguments. Infinitely many of the numbers ζ(2n + 1) must be irrational, and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.

See also

• Riemann zeta function
• Basel problem — ζ(2)
• List of sums of reciprocals

References

• Amdeberhan, Tewodros (1996), "Faster and faster convergent series for $\zeta (3)$", El. J. Combinat., 3 (1).
• Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", El. J. Combinat., 4 (2), arXiv:math/9804121, Bibcode:1998math......4121A.
• Apéry, Roger (1979), "Irrationalité de $\zeta 2$ et $\zeta 3$", Astérisque, 61: 11–13.
• Berndt, Bruce C. (1989), Ramanujan's notebooks, Part II, Springer.
• Beukers, F. (1979), "A Note on the Irrationality of $\zeta (2)$ and $\zeta (3)$", Bull. London Math. Soc., 11 (3): 268–272, doi:10.1112/blms/11.3.268.
• Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results", The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5, S2CID 120943474.
• Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta (3)$ and $\zeta (5)$, arXiv:math.CA/9803067.
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• Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: $\zeta (3)$.
• Hjortnaes, M. M. (August 1953), Overføring av rekken $\sum _{k=1}^{\infty }\left({\frac {1}{k^{3}}}\right)$ til et bestemt integral, in Proc. 12th Scandinavian Mathematical Congress, Lund, Sweden: Scandinavian Mathematical Society, pp. 211–213.
• Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347.
• Markov, A. A. (1890), "Mémoire sur la transformation des séries peu convergentes en séries très convergentes", Mém. De l'Acad. Imp. Sci. De St. Pétersbourg, t. XXXVII, No. 9: 18pp.
• Mohammed, Mohamud (2005), "Infinite families of accelerated series for some classical constants by the Markov-WZ method", Discrete Mathematics and Theoretical Computer Science, 7: 11–24, doi:10.46298/dmtcs.342.
• Mollin, Richard A. (2009), Advanced Number Theory with Applications, Discrete Mathematics and Its Applications, CRC Press, p. 220, ISBN 9781420083293.
• Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II.
• Rivoal, Tanguy (2000), "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs", Comptes Rendus de l'Académie des Sciences, Série I, 331 (4): 267–270, arXiv:math/0008051, Bibcode:2000CRASM.331..267R, doi:10.1016/S0764-4442(00)01624-4, S2CID 119678120.
• Srivastava, H. M. (December 2000), "Some Families of Rapidly Convergent Series Representations for the Zeta Functions" (PDF), Taiwanese Journal of Mathematics, 4 (4): 569–599, doi:10.11650/twjm/1500407293, OCLC 36978119, retrieved 2015-08-22.
• van der Poorten, Alfred (1979), "A proof that Euler missed ... Apéry's proof of the irrationality of $\zeta (3)$" (PDF), The Mathematical Intelligencer, 1 (4): 195–203, doi:10.1007/BF03028234, S2CID 121589323, archived from the original (PDF) on 2011-07-06.
• Wedeniwski, Sebastian (2001), Simon Plouffe (ed.), The Value of Zeta(3) to 1,000,000 places, Project Gutenberg (Message to Simon Plouffe, with all decimal places but a shorter text edited by Simon Plouffe).
• Wedeniwski, Sebastian (13 December 1998), The Value of Zeta(3) to 1,000,000 places (Message to Simon Plouffe, with original text but only some decimal places).
• Yee, Alexander J. (2009), Large Computations.
• Yee, Alexander J. (2017), Zeta(3) - Apéry's Constant

• Nag, Dipanjan (2015), Calculated Apéry's constant to 400,000,000,000 Digit, A world record
• Zudilin, Wadim (2001), "One of the numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, $\zeta (11)$ is irrational", Russ. Math. Surv., 56 (4): 774–776, Bibcode:2001RuMaS..56..774Z, doi:10.1070/RM2001v056n04ABEH000427, S2CID 250734661.
• Zudilin, Wadim (2002), An elementary proof of Apéry's theorem, arXiv:math/0202159, Bibcode:2002math......2159Z.

Further reading

• Ramaswami, V. (1934), "Notes on Riemann's $\zeta$-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. ISBN 978-0-691-22759-7. OCLC 1260168397.{{cite book}}: CS1 maint: location missing publisher (link)

External links

• Weisstein, Eric W., "Apéry's constant", MathWorld
• 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.

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