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%I M0020 #401 Apr 16 2024 13:49:03
%S 1,2,0,2,0,5,6,9,0,3,1,5,9,5,9,4,2,8,5,3,9,9,7,3,8,1,6,1,5,1,1,4,4,9,
%T 9,9,0,7,6,4,9,8,6,2,9,2,3,4,0,4,9,8,8,8,1,7,9,2,2,7,1,5,5,5,3,4,1,8,
%U 3,8,2,0,5,7,8,6,3,1,3,0,9,0,1,8,6,4,5,5,8,7,3,6,0,9,3,3,5,2,5,8,1,4,6,1,9,9,1,5
%N Apéry's number or Apéry's constant zeta(3). Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3.
%C Sometimes called Apéry's constant.
%C "A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apéry succeeded in proving that Zeta(3) is irrational. In Chapter 8 we pointed out that the probability that two random integers are relatively prime is 6/Pi^2, which is 1/Zeta(2). This generalizes to: The probability that k random integers are relatively prime is 1/Zeta(k) ... ." [Stan Wagon]
%C In 2001 Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See the Rivoal link for further information and references.
%C The reciprocal of this constant is the probability that three integers chosen randomly using uniform distribution are relatively prime. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2005
%C Also the value of zeta(1,2), the double zeta-function of arguments 1 and 2. - _R. J. Mathar_, Oct 10 2011
%C Also the length of minimal spanning tree for large complete graph with uniform random edge lengths between 0 and 1, cf. link to John Baez's comment. - _M. F. Hasler_, Sep 26 2017
%C Sum of the inverses of the cubes (A000578). - _Michael B. Porter_, Nov 27 2017
%C This number is the average value of sigma_2(n)/n^2 where sigma_2(n) is the sum of the squares of the divisors of n. - _Dimitri Papadopoulos_, Jan 07 2022
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 40-53.
%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.
%D R. William Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, Computers in Mathematics (Stanford CA, 1986); Lecture Notes in Pure and Appl. Math., Dekker, New York, 125 (1990), 261-284; MR 91h:11154.
%D Xavier Gourdon, Analyse, Les Maths en tête, Ellipses, 1994, Exemple 3, page 224.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F17, Series associated with the zeta-function, p. 391.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press; 6 edition (2008), pp. 47, 268-269.
%D Paul Levrie, The Ubiquitous Apéry Number, Math. Intelligencer, Vol. 45, No. 2, 2023, pp. 118-119.
%D A. A. Markoff, Mémoire sur la transformation de séries peu convergentes en séries très convergentes, Mém. de l'Acad. Imp. Sci. de St. Pétersbourg, XXXVII, 1890.
%D Paul J. Nahin, In Pursuit of Zeta-3, Princeton University Press, 2021.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Stan Wagon, Mathematica In Action, W. H. Freeman and Company, NY, 1991, page 354.
%D A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Dover (1987), Ex. 92-93.
%H Harry J. Smith, <a href="/A002117/b002117.txt">Table of n, a(n) for n = 1..20002</a>
%H T. Amdeberhan, <a href="https://arxiv.org/abs/math/9804126">Faster and Faster convergent series for zeta(3)</a>, arXiv:math/9804126 [math.CO], 1998.
%H Kunihiro Aoki and Ryo Furue, <a href="https://arxiv.org/abs/2103.10221">A model for the size distribution of marine microplastics: a statistical mechanics approach</a>, arXiv:2103.10221 [physics.ao-ph], 2021.
%H Peter Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>.
%H Peter Bala, <a href="/A002117/a002117.rtf">Some series for zeta(3)</a>, Nov 2023.
%H John Baez, <a href="https://johncarlosbaez.wordpress.com/2017/08/08/applied-algebraic-topology-2017/#comment-97126">Comments about zeta(3)</a>, Azimuth Project blog, August 2017.
%H J. Borwein and D. Bradley, <a href="https://arxiv.org/abs/math/0505124">Empirically determined Apéry-like formulas for zeta(4n+3)</a>, arXiv:math/0505124 [math.CA], 2005.
%H Mainendra Kumar Dewangan and Subhra Datta, <a href="https://doi.org/10.1017/jfm.2020.134">Effective permeability tensor of confined flows with wall grooves of arbitrary shape</a>, J. of Fluid Mechanics (2020) Vol. 891.
%H Dr. Math, <a href="https://web.archive.org/web/20201111172246/http://mathforum.org/library/drmath/view/55801.html">Probability of Random Numbers Being Coprime</a>.
%H L. Euler, <a href="https://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, arXiv:math/0506415 [math.HO], 2005-2008.
%H L. Euler, <a href="http://eulerarchive.maa.org/pages/E041.html">De summis serierum reciprocarum</a>, E41.
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Zeta3/zeta3.html">The Apery's constant: zeta(3)</a>
%H Brady Haran and Tony Padilla, <a href="https://www.youtube.com/watch?v=ur-iLy4z3QE">Apéry's constant (calculated with Twitter)</a>, Numberphile video (2017).
%H W. Janous, <a href="http://www.emis.de/journals/JIPAM/article652.html?sid=652">Around Apéry's constant</a>, J. Inequ. Pure Appl. Math. 7(1) (2006), #35.
%H Yasuyuki Kachi and Pavlos Tzermias, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Tzermias/tzermias2.html">Infinite products involving zeta(3) and Catalan's constant</a>, Journal of Integer Sequences, 15 (2012), #12.9.4.
%H Masato Kobayashi, <a href="https://arxiv.org/abs/2108.01247">Integral representations for zeta(3) with the inverse sine function</a>, arXiv:2108.01247 [math.NT], 2021.
%H M. Kondratiewa and S. Sadov, <a href="https://arxiv.org/abs/math/0405592">Markov's transformation of series and the WZ method</a>, arXiv:math/0405592 [math.CA], 2004.
%H Tobias Kyrion, <a href="https://arxiv.org/abs/2008.05573">A closed-form expression for zeta(3)</a>, arXiv:2008.05573 [math.GM], 2020.
%H John Landen, <a href="https://archive.org/details/mathematicalmem00landgoog/page/n130/mode/2up">Mathematical memoirs respecting a variety of subjects Vol. I</a>, London, 1780.
%H F. M. S. Lima, <a href="http://arxiv.org/abs/0910.2684">Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study</a>, arXiv:0910.2684 [math.NT], 2009-2012.
%H Jonah Lissner, <a href="https://www.researchgate.net/publication/353322808_Theoretical_Physics_Utilizations_Of_Riemann_Zeta_Function_Odd_Positive_Integer_Three">Theoretical Physics Utilizations Of Riemann Zeta Function Odd Positive Integer Three</a>, ResearchGate (2024).
%H C. Lupu and D. Orr, <a href="https://doi.org/10.1007/s11139-018-0081-0">Series representations for the Apéry constant zeta(3) involving the values zeta(2n)</a>, Ramanujan J. 48(3) (2019), 477-494.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1207.5845">Yet another table of integrals</a>, arXiv:1207.5845 [math.CA], 2012-2014.
%H G. P. Michon, <a href="http://www.numericana.com/fame/apery.htm">Roger Apéry</a>, Numericana.
%H S. D. Miller, <a href="https://web.archive.org/web/20070614030202/https://www.math.princeton.edu/mathlab/book/papers/simplerzeta3SDMiller.pdf">An Easier Way to Show zeta(3) is Irrational</a>.
%H Simon Plouffe, <a href="https://web.archive.org/web/20080205213257/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap97.html">Zeta(3) or Apéry's constant to 2000 places</a>.
%H Simon Plouffe, <a href="/A293904/a293904_4096.gz">Zeta(2) to Zeta(4096) to 2048 digits each</a> (gzipped file).
%H A. van der Poorten, <a href="https://web.archive.org/web/20110415154621/http://www.ift.uni.wroc.pl/~mwolf/Poorten_MI_195_0.pdf">A Proof that Euler Missed</a>.
%H Tanguy Rivoal, <a href="http://algo.inria.fr/seminars/sem01-02/rivoal.ps">Irrationality of the zeta Function on Odd Integers</a> [ps file].
%H Tanguy Rivoal, <a href="/A002117/a002117_3.pdf">Irrationality of the zeta Function on Odd Integers</a> [pdf file].
%H Ernst E. Scheufens, <a href="https://www.jstor.org/stable/10.4169/math.mag.84.1.026">From Fourier series to rapidly convergent series for zeta(3)</a>, Mathematics Magazine, Vol. 84, No. 1 (2011), pp. 26-32.
%H G. Villemin's Almanach of Numbers, <a href="https://diconombre.pagesperso-orange.fr/UnP2.htm#Ap%C3%A9ry">Constante d'Apéry</a> (in French).
%H S. Wedeniwski, <a href="https://web.archive.org/web/20050205150332/http://www.gutenberg.org/dirs/etext01/zeta310.txt">The value of zeta(3) to 1000000 places</a> [Gutenberg Project Etext].
%H S. Wedeniwski, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/Zeta3.txt">Apery's constant to 128000026 decimal digits</a>.
%H S. Wedeniwski, <a href="https://web.archive.org/web/20040328200336/http://ftp.ibiblio.org/pub/docs/books/gutenberg/etext01/zeta310.txt">The value of zeta(3) to 1000000 decimal digits</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AperysConstant.html">Apéry's Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RelativelyPrime.html">Relatively Prime</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>.
%H H. Wilf, <a href="http://www.emis.de/journals/DMTCS/volumes/abstracts/dm030406.abs.html">Accelerated series for universal constants, by the WZ method</a>, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.
%H J. W. Wrench, Jr., <a href="/A002117/a002117_1.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a>
%H Wenzhe Yang, <a href="https://arxiv.org/abs/1911.02608">Apéry's irrationality proof, mirror symmetry and Beukers' modular forms</a>, arXiv:1911.02608 [math.NT], 2019.
%H Wadim Zudilin, <a href="http://arXiv.org/abs/math/0202159">An elementary proof of Apéry's theorem</a>, arXiv:math/0202159 [math.NT], 2002.
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F Lima gives an approximation to zeta(3) as (236*log(2)^3)/197 - 283/394*Pi*log(2)^2 + 11/394*Pi^2*log(2) + 209/394*log(sqrt(2) + 1)^3 - 5/197 + (93*Catalan*Pi)/197. - _Jonathan Vos Post_, Oct 14 2009 [Corrected by _Wouter Meeussen_, Apr 04 2010]
%F zeta(3) = 5/2*Integral_(x=0..2*log((1+sqrt(5))/2), x^2/(exp(x)-1)) + 10/3*(log((1+sqrt(5))/2))^3. - _Seiichi Kirikami_, Aug 12 2011
%F zeta(3) = -4/3*Integral_{x=0..1} log(x)/x*log(1+x) = Integral_{x=0..1} log(x)/x*log(1-x) = -4/7*Integral_{x=0..1} log(x)/x*log((1+x)/(1-x)) = 4*Integral_{x=0..1} 1/x*log(1+x)^2 = 1/2*Integral_{x=0..1} 1/x*log(1-x)^2 = -16/7*Integral_{x=0..Pi/2} x*log(2*cos(x)) = -4/Pi*Integral_{x=0..Pi/2} x^2*log(2*cos(x)). - _Jean-François Alcover_, Apr 02 2013, after _R. J. Mathar_
%F From _Peter Bala_, Dec 04 2013: (Start)
%F zeta(3) = (16/7)*Sum_{k even} (k^3 + k^5)/(k^2 - 1)^4.
%F zeta(3) - 1 = Sum_{k >= 1} 1/(k^3 + 4*k^7) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 - ... - (n - 1)^6/((2*n - 1)*(n^2 - n + 5) - ...))))) (continued fraction).
%F More generally, there is a sequence of polynomials P(n,x) (of degree 2*n) such that
%F zeta(3) - Sum_{k = 1..n} 1/k^3 = Sum_{k >= 1} 1/( k^3*P(n,k-1)*P(n,k) ) = 1/((2*n^2 + 2*n + 1) - 1^6/(3*(2*n^2 + 2*n + 3) - 2^6/(5*(2*n^2 + 2*n + 7) - 3^6/(7*(2*n^2 + 2*n + 13) - ...)))) (continued fraction). See A143003 and A143007 for details.
%F Series acceleration formulas:
%F zeta(3) = (5/2)*Sum_{n >= 1} (-1)^(n+1)/( n^3*binomial(2*n,n) )
%F = (5/2)*Sum_{n >= 1} P(n)/( (2*n(2*n - 1))^3*binomial(4*n,2*n) )
%F = (5/2)*Sum_{n >= 1} (-1)^(n+1)*Q(n)/( (3*n(3*n - 1)*(3*n - 2))^3*binomial(6*n,3*n) ), where P(n) = 24*n^3 + 4*n^2 - 6*n + 1 and Q(n) = 9477*n^6 - 11421*n^5 + 5265*n^4 - 1701*n^3 + 558*n^2 - 108*n + 8 (Bala, section 7). (End)
%F zeta(3) = Sum_{n >= 1} (A010052(n)/n^(3/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(3/2) ). - _Mikael Aaltonen_, Feb 22 2015
%F zeta(3) = Product_{k>=1} 1/(1 - 1/prime(k)^3). - _Vaclav Kotesovec_, Apr 30 2020
%F zeta(3) = 4*(2*log(2) - 1 - 2*Sum_{k>=2} zeta(2*k+1)/2^(2*k+1)). - _Jorge Coveiro_, Jun 21 2020
%F zeta(3) = (4*zeta'''(1/2)*(zeta(1/2))^2-12*zeta(1/2)*zeta'(1/2)*zeta''(1/2)+8*(zeta'(1/2))^3-Pi^3*(zeta(1/2))^3)/(28*(zeta(1/2))^3). - _Artur Jasinski_, Jun 27 2020
%F zeta(3) = Sum_{k>=1} H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - _Amiram Eldar_, Jul 31 2020
%F From _Artur Jasinski_, Sep 30 2020: (Start)
%F zeta(3) = (5/4)*Li_3(1/f^2) + Pi^2*log(f)/6 - 5*log(f)^3/6,
%F zeta(3) = (8/7)*Li_3(1/2) + (2/21)*Pi^2 log(2) - (4/21) log(2)^3, where f is golden ratio (A001622) and Li_3 is the polylogarithm function, formulas published by John Landen in 1780, p. 118. (End)
%F zeta(3) = (1/2)*Integral_{x=0..oo} x^2/(e^x-1) dx (Gourdon). - _Bernard Schott_, Apr 28 2021
%F From _Peter Bala_, Jan 18 2022: (Start)
%F zeta(3) = 1 + Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)) = 25/24 + (2!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)) = 28333/27000 + (3!)^4*(Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)*(4*n^4 + 3^4)). In general, for k >= 1, we have zeta(3) = r(k) + (k!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*...*(4*n^4 + k^4)), where r(k) is rational.
%F zeta(3) = (6/7) + (64/7)*Sum_{n >= 1} n/(4*n^2 - 1)^3.
%F More generally, for k >= 0, it appears that zeta(3) = a(k) + b(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^3, where a(k) and b(k) are rational.
%F zeta(3) = (10/7) - (128/7)*Sum_{n >= 1} n/(4*n^2 - 1)^4.
%F More generally, for k >= 0, it appears that zeta(3) = c(k) + d(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^4, where c(k) and d(k) are rational. [added Nov 27 2023: for the values of a(k), b(k), c(k) and d(k) see the Bala 2023 link, Sections 8 and 9.]
%F zeta(3) = 2/3 + (2^13)/(3*7)*Sum_{n >= 1} n^3/(4*n^2 - 1)^6. (End)
%F zeta(3) = -Psi(2)(1/2)/14 (the second derivative of digamma function evaluated at 1/2). - _Artur Jasinski_, Mar 18 2022
%F zeta(3) = -(8*Pi^2/9) * Sum_{k>=0} zeta(2*k)/((2*k+1)*(2*k+3)*4^k) = (2*Pi^2/9) * (log(2) + 2 * Sum_{k>=0} zeta(2*k)/((2*k+3)*4^k)) (Scheufens, 2011). - _Amiram Eldar_, May 28 2022
%F zeta(3) = Sum_{k>=1} (30*k-11) / (4*(2k-1)*k^3*(binomial(2k,k))^2) (Gosper, 1986 and Richard K. Guy reference). - _Bernard Schott_, Jul 20 2022
%F zeta(3) = (4/3)*Integral_{x >= 1} x*log(x)*(1 + log(x))*log(1 + 1/x^x) dx = (2/3)*Integral_{x >= 1} x^2*log(x)^2*(1 + log(x))/(1 + x^x) dx. - _Peter Bala_, Nov 27 2023
%F zeta_3(n) = 1/180*(-360*n^3*f(-3, n/4) + Pi^3*(n^4 + 20*n^2 + 16))/(n*(n^2 + 4)), where f(-3, n) = Sum_{k>=1} 1/(k^3*(exp(Pi*k/n) - 1)). Will give at least 1 digit of precision/term, example: zeta_3(5) = 1.202056944732.... - _Simon Plouffe_, Dec 21 2023
%F zeat(3) = 1 + (1/2)*Sum_{n >= 1} (2*n + 1)/(n^3*(n + 1)^3) = 5/4 - (1/4)*Sum_{n >= 1} (2*n + 1)/(n^4*(n + 1)^4) = 147/120 + (2/15)*Sum_{n >= 1} (2*n + 1)/(n^5*(n + 1)^5) - (64/15)*Sum_{n >= 1} (n + 1)/(n^5*(n + 2)^5) = 19/16 + (128/21)*Sum_{n >= 1} (n + 1)/(n^6*(n + 2)^6) - (1/21)*Sum_{n >= 1} (2*n + 1)/(n^6*(n + 1)^6). - _Peter Bala_, Apr 15 2024
%e 1.2020569031595942853997...
%p # Calculates an approximation with n exact decimal places (small deviation
%p # in the last digits are possible). Goes back to ideas of A. A. Markoff 1890.
%p zeta3 := proc(n) local s, w, v, k; s := 0; w := -1; v := 4;
%p for k from 2 by 2 to 7*n/2 do
%p w := -w*v/k;
%p v := v + 8;
%p s := s + 1/(w*k^3);
%p od; 20*s; evalf(%, n) end:
%p zeta3(10000); # _Peter Luschny_, Jun 10 2020
%t RealDigits[ N[ Zeta[3], 100] ] [ [1] ]
%t (* Second program (historical interest): *)
%t d[n_] := 34*n^3 + 51*n^2 + 27*n + 5; 6/Fold[Function[d[#2-1] - #2^6/#1], 5, Reverse[Range[100]]] // N[#, 108]& // RealDigits // First
%t (* _Jean-François Alcover_, Sep 19 2014, after Apéry's continued fraction *)
%o (PARI) default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002117.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 19 2009
%o (Maxima) fpprec : 100$ ev(bfloat(zeta(3)))$ bfloat(%); /* _Martin Ettl_, Oct 21 2012 */
%o (Python)
%o from mpmath import mp, apery
%o mp.dps=109
%o print([int(z) for z in list(str(apery).replace('.', ''))[:-1]]) # _Indranil Ghosh_, Jul 08 2017
%o (Magma) L:=RiemannZeta(: Precision:=100); Evaluate(L,3); // _G. C. Greubel_, Aug 21 2018
%Y Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, A013677, A059956 (6/Pi^2), A084225; A084226.
%Y Cf. A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8; A233091: 7*zeta(3)/8.
%Y Cf. A001008, A002805, A143003, A143007.
%Y Cf. A000578 (cubes).
%Y Cf. sums of inverses: A152623 (tetrahedral numbers), A175577 (octahedral numbers), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).
%K cons,nonn,nice,changed
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
%E Additional comments from _Robert G. Wilson v_, Dec 08 2000
%E Quotation from Stan Wagon corrected by _N. J. A. Sloane_ on Dec 24 2005. Thanks to Jose Brox for noticing this error.
%E Edited by _M. F. Hasler_, Sep 26 2017
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