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A002117 Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3.
(Formerly M0020)
224

%I M0020

%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 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' comment. - _M. F. Hasler_, Sep 26 2017

%C Sum of the inverses of the cubes (A000578). - _Michael B. Porter_, Nov 27 2017

%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 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 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 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 Dr. Math, <a href="http://mathforum.org/library/drmath/view/55801.html">Probability of Random Numbers Being Coprime</a>.

%H P. Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>.

%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 Nelson A. Carella, <a href="https://arxiv.org/abs/1906.10618">The Zeta Quotient Zeta(3)/Pi^3 is Irrational</a>, arXiv:1906.10618 [math.GM], 2019.

%H N. A. Carella, <a href="https://arxiv.org/abs/2003.01532">Irrationality Exponents For Even Zeta Constants</a>, arXiv:2003.01532 [math.GM], 2020.

%H Mainendra Kumar Dewangan, 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 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 Apery'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 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">A simple approximate expression for the Ape'ry's constant accurate to 21 digits</a>, arXiv:0910.2684 [math.NT], 2009-2012.

%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 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">Apery'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 Apery'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_, Fri 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)

%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

%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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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)