Apéry's constant
Template:Infobox non-integer number
In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number
- [math]\displaystyle{ \begin{align} \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{align} }[/math]
where ζ is the Riemann zeta function. It has an approximate value of[1]
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[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.
Irrational number
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Unsolved problem in mathematics: Is Apéry's constant transcendental? (more unsolved problems in mathematics)
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ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.[3] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[4] and simpler proofs were found later.[5]
Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3),
- [math]\displaystyle{ \zeta(3) = \int_0^1 \int_0^1 \int_0^1 \frac{1}{1-xyz}\, dx\, dy\, dz, }[/math]
by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that
- [math]\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, }[/math]
where [math]\displaystyle{ |I| \leq \zeta(3) (1-\sqrt{2})^{4n} }[/math], [math]\displaystyle{ P_n(z) }[/math] are the Legendre polynomials, and the subsequences [math]\displaystyle{ b_n, 2 \operatorname{lcm}(1,2,\ldots,n) \cdot a_n \in \mathbb{Z} }[/math] 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:
- [math]\displaystyle{ \zeta(3) = \sum_{k=1}^\infty \frac{1}{k^3}, }[/math]
Leonhard Euler gave the series representation:[6]
- [math]\displaystyle{ \zeta(3) = \frac{\pi^2}{7} \left(1 - 4\sum_{k=1}^\infty \frac{\zeta (2k)}{2^{2k}(2k + 1)(2k + 2)}\right) }[/math]
in 1772, which was subsequently rediscovered several times.[7]
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,[8] rediscovered by Hjortnaes in 1953,[9] and rediscovered once more and widely advertised by Apéry in 1979:[3]
- [math]\displaystyle{ \zeta(3) = \frac{5}{2} \sum_{k=1}^\infty (-1)^{k-1} \frac{k!^2}{(2k)! k^3}. }[/math]
The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[10]
- [math]\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)!}. }[/math]
The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[11]
- [math]\displaystyle{ \zeta(3) = \frac{1}{64} \sum_{k=0}^\infty (-1)^k \frac{k!^{10} (205k^2 + 250k + 77)}{(2k + 1)!^5}. }[/math]
The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[12]
- [math]\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}. }[/math]
It has been used to calculate Apéry's constant with several million correct decimal places.[13]
The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[14]
- [math]\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}. }[/math]
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.[15]
Thue-Morse sequence
The following representation was found by Tóth in 2022:[16]
- [math]\displaystyle{ \begin{align} \sum_{n\geq1} \frac{9 t_{n-1} + 7 t_n}{n^3} &= 8 \zeta(3),\end{align} }[/math]
where [math]\displaystyle{ (t_n)_{n\geq0} }[/math] is the [math]\displaystyle{ n^{\rm th} }[/math] term of the Thue-Morse sequence. In fact, this is a special case of the following formula (valid for all [math]\displaystyle{ s }[/math] with real part greater than [math]\displaystyle{ 1 }[/math]):
- [math]\displaystyle{ (2^s+1) \sum_{n\geq1} \frac{t_{n-1}}{n^s} + (2^s-1) \sum_{n\geq1} \frac{t_{n}}{n^s} = 2^s \zeta(s). }[/math]
Others
The following series representation was found by Ramanujan:[17]
- [math]\displaystyle{ \zeta(3) = \frac{7}{180} \pi^3 - 2 \sum_{k=1}^\infty \frac{1}{k^3 (e^{2\pi k} - 1)}. }[/math]
The following series representation was found by Simon Plouffe in 1998:[18]
- [math]\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)}. }[/math]
(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[19]
- [math]\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 }[/math]
and[20]
- [math]\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. }[/math]
Also,[21]
- [math]\displaystyle{ \begin{align} \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{align} }[/math]
A connection to the derivatives of the gamma function[22]
- [math]\displaystyle{ \zeta(3) = -\tfrac{1}{2}(\Gamma'''(1) + \gamma^3+ \tfrac{1}{2}\pi^2\gamma) = -\tfrac{1}{2} \psi^{(2)}(1) }[/math]
is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.[23]
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[1] September 2001 200001000 Shigeru Kondo & Xavier Gourdon February 2002 600001000 Shigeru Kondo & Xavier Gourdon February 2003 1000000000 Patrick Demichel & Xavier Gourdon[24] April 2006 10000000000 Shigeru Kondo & Steve Pagliarulo January 21, 2009 15510000000 Alexander J. Yee & Raymond Chan[25] February 15, 2009 31026000000 Alexander J. Yee & Raymond Chan[25] September 17, 2010 100000001000 Alexander J. Yee[26] September 23, 2013 200000001000 Robert J. Setti[26] August 7, 2015 250000000000 Ron Watkins[26] December 21, 2015 400000000000 Dipanjan Nag[27] August 13, 2017 500000000000 Ron Watkins[26] May 26, 2019 1000000000000 Ian Cutress[28] July 26, 2020 1200000000100 Seungmin Kim[29][30]
Reciprocal
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).[31]) 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).[31])
Extension to ζ(2n + 1)
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,[32] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[33]
See also
Notes
- ↑ 1.0 1.1 Wedeniwski (2001).
- ↑ Frieze (1985).
- ↑ 3.0 3.1 Apéry (1979).
- ↑ van der Poorten (1979).
- ↑ (Beukers 1979); (Zudilin 2002).
- ↑ Euler (1773).
- ↑ Srivastava (2000), p. 571 (1.11).
- ↑ Markov (1890).
- ↑ Hjortnaes (1953).
- ↑ Amdeberhan (1996).
- ↑ Amdeberhan & Zeilberger (1997).
- ↑ (Wedeniwski 1998); (Wedeniwski 2001). In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from (Amdeberhan Zeilberger). The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).
- ↑ (Wedeniwski 1998); (Wedeniwski 2001).
- ↑ Mohammed (2005).
- ↑ Broadhurst (1998).
- ↑ Tóth, László (2022). "Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence". Integers 22 (article 98). http://math.colgate.edu/~integers/w98/w98.pdf.
- ↑ (Berndt 1989).
- ↑ Plouffe (1998).
- ↑ Jensen (1895).
- ↑ Beukers (1979).
- ↑ Blagouchine (2014).
- ↑ https://scipp.ucsc.edu/~haber/archives/physics116A10/psifun_10.pdf
- ↑ Evgrafov et al. (1969), exercise 30.10.1.
- ↑ Gourdon & Sebah (2003).
- ↑ 25.0 25.1 Yee (2009).
- ↑ 26.0 26.1 26.2 26.3 Yee (2017).
- ↑ Nag (2015).
- ↑ Records set by y-cruncher, http://www.numberworld.org/y-cruncher/records.html, retrieved June 8, 2019.
- ↑ Records set by y-cruncher, http://www.numberworld.org/y-cruncher/, retrieved August 10, 2020.
- ↑ Apéry's constant world record by Seungmin Kim, 28 July 2020, https://ehfd.github.io/world-record/aperys-constant/, retrieved July 28, 2020.
- ↑ 31.0 31.1 Mollin (2009).
- ↑ Rivoal (2000).
- ↑ Zudilin (2001).
References
- Amdeberhan, Tewodros (1996), "Faster and faster convergent series for [math]\displaystyle{ \zeta(3) }[/math]", El. J. Combinat. 3 (1), http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r13.
- Amdeberhan, Tewodros; Zeilberger, Doron (1997), "Hypergeometric Series Acceleration Via the WZ method", El. J. Combinat. 4 (2), Bibcode: 1998math......4121A, http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r3.
- Apéry, Roger (1979), "Irrationalité de [math]\displaystyle{ \zeta 2 }[/math] et [math]\displaystyle{ \zeta 3 }[/math]", Astérisque 61: 11–13, http://www.numdam.org/item/AST_1979__61__11_0/.
- Berndt, Bruce C. (1989), Ramanujan's notebooks, Part II, Springer.
- Beukers, F. (1979), "A Note on the Irrationality of [math]\displaystyle{ \zeta(2) }[/math] and [math]\displaystyle{ \zeta(3) }[/math]", 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.
- Broadhurst, D.J. (1998), Polylogarithmic ladders, hypergeometric series and the ten millionth digits of [math]\displaystyle{ \zeta(3) }[/math] and [math]\displaystyle{ \zeta(5) }[/math].
- Euler, Leonhard (1773), "Exercitationes analyticae" (in la), Novi Commentarii Academiae Scientiarum Petropolitanae 17: 173–204, http://math.dartmouth.edu/~euler/docs/originals/E432.pdf, retrieved 2008-05-18.
- Evgrafov, M. A.; Bezhanov, K. A.; Sidorov, Y. V.; Fedoriuk, M. V.; Shabunin, M. I. (1969), A Collection of Problems in the Theory of Analytic Functions [in Russian], Moscow: Nauka.
- "On the value of a random minimum spanning tree problem", Discrete Applied Mathematics 10 (1): 47–56, 1985, doi:10.1016/0166-218X(85)90058-7.
- Gourdon, Xavier; Sebah, Pascal (2003), The Apéry's constant: [math]\displaystyle{ \zeta(3) }[/math], http://numbers.computation.free.fr/Constants/Zeta3/zeta3.html.
- Hjortnaes, M. M. (August 1953), Overføring av rekken [math]\displaystyle{ \sum_{k=1}^\infty\left(\frac{1}{k^3}\right) }[/math] 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, https://books.google.com/books?id=6I1setlljDYC&pg=PA220.
- Plouffe, Simon (1998), Identities inspired from Ramanujan Notebooks II, http://www.lacim.uqam.ca/~plouffe/identities.html.
- 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, doi:10.1016/S0764-4442(00)01624-4, Bibcode: 2000CRASM.331..267R.
- Srivastava, H. M. (December 2000), "Some Families of Rapidly Convergent Series Representations for the Zeta Functions", Taiwanese Journal of Mathematics 4 (4): 569–599, doi:10.11650/twjm/1500407293, OCLC 36978119, http://society.math.ntu.edu.tw/~journal/tjm/V4N4/tjm0012_3.pdf, retrieved 2015-08-22.
- van der Poorten, Alfred (1979), "A proof that Euler missed ... Apéry's proof of the irrationality of [math]\displaystyle{ \zeta(3) }[/math]", The Mathematical Intelligencer 1 (4): 195–203, doi:10.1007/BF03028234, http://www.maths.mq.edu.au/~alf/45.pdf.
- Wedeniwski, Sebastian (2001), Simon Plouffe, ed., The Value of Zeta(3) to 1,000,000 places, Project Gutenberg, http://www.gutenberg.org/cache/epub/2583/pg2583.html (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, http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/Zeta3.txt (Message to Simon Plouffe, with original text but only some decimal places).
- Yee, Alexander J. (2009), Large Computations, http://www.numberworld.org/nagisa_runs/computations.html.
- Yee, Alexander J. (2017), Zeta(3) - Apéry's Constant, http://www.numberworld.org/digits/Zeta%283%29/
- Nag, Dipanjan (2015), Calculated Apéry's constant to 400,000,000,000 Digit, A world record, https://dipanjan.me/calculated-aperysconstant-upto-400000000000-digit-a-world-record/
- Zudilin, Wadim (2001), "One of the numbers [math]\displaystyle{ \zeta(5) }[/math], [math]\displaystyle{ \zeta(7) }[/math], [math]\displaystyle{ \zeta(9) }[/math], [math]\displaystyle{ \zeta(11) }[/math] is irrational", Russ. Math. Surv. 56 (4): 774–776, doi:10.1070/RM2001v056n04ABEH000427, Bibcode: 2001RuMaS..56..774Z.
- Zudilin, Wadim (2002), An elementary proof of Apéry's theorem, Bibcode: 2002math......2159Z.
Further reading
- Ramaswami, V. (1934), "Notes on Riemann's [math]\displaystyle{ \zeta }[/math]-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.
External links
- Weisstein, Eric W.. "Apéry's constant". http://mathworld.wolfram.com/AperysConstant.html.
- Plouffe, Simon, Zeta(3) or Apéry constant to 2000 places, http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap97.html, retrieved 2005-07-29
- Setti, Robert J. (2015), Apéry's Constant - Zeta(3) - 200 Billion Digits, http://settifinancial.com/01042-aperys-constant-zeta3-world-record-computation/.
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