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%I A002117 M0020
%S A002117 1,2,0,2,0,5,6,9,0,3,1,5,9,5,9,4,2,8,5,3,9,9,7,3,8,1,6,1,5,1,1,4,4,9,9,
               9,
%T A002117 0,7,6,4,9,8,6,2,9,2,3,4,0,4,9,8,8,8,1,7,9,2,2,7,1,5,5,5,3,4,1,8,3,8,2,
               0,
%U A002117 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 A002117 Decimal expansion of zeta(3) = sum_{m=1 .. infinity} 1/m^3.
%C A002117 Sometimes called Apery's constant.
%C A002117 "A natural question is whether Zeta(3) is a rational multiple of Pi^3. 
               This is not known, though in 1978 R. Apery 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 A002117 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 A002117 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
%D A002117 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, pp. 40-53
%D A002117 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 A002117 Hardy and Wright, 'An Introduction to the Theory of Numbers' pp. 47,268-269
%D A002117 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002117 Stan Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, 
               page 354.
%D A002117 Yaglom and Yaglom, 'Challenging Mathematical Problems with Elementary 
               Solutions' ex. 92-93
%H A002117 Harry J. Smith, <a href="b002117.txt">Table of n, a(n) for n=1,...,20002</
               a>
%H A002117 T. Amdeberhan, <a href="http://arXiv.org/abs/math.CO/9804126">Faster 
               and Faster convergent series for zeta(3)</a>
%H A002117 Author?, <a href="http://mathforum.org/library/drmath/view/55801.html">
               Probability of Random Numbers Being Coprime</a>
%H A002117 Author?, <a href="http://www.ballandclaw.com/upi/coprime.html">Probability 
               of two numbers being coprime</a>
%H A002117 J. Borwein and D. Bradley, <a href="http://arXiv.org/abs/math.CA/0505124">
               Empirically determined Ap'ery-like formulae for zeta(4n+3)</a>
%H A002117 L. Euler, <a href="http://arXiv.org/abs/math.HO/0506415">On the sums 
               of series of reciprocals</a>
%H A002117 L. Euler, <a href="http://www.eulerarchive.org">De summis serierum reciprocarum</
               a>, E41.
%H A002117 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/
               Constants/Zeta3/zeta3.html">The Apery's constant:zeta(3)</a>
%H A002117 W. Janous, <a href="http://jipam.vu.edu.au/article.php?sid=652">Around 
               Apery's constant</a>, J. Inequ. Pure Appl. Math. 7 (2006) vol. 1, 
               #35
%H A002117 M. Kondratiewa and S. Sadov, <a href="http://arXiv.org/abs/math.CA/0405592">
               Markov's transformation of series and the WZ method</a>
%H A002117 S. D. Miller, <a href="http://www.math.princeton.edu/mathlab/book/papers/
               simplerzeta3SDMiller.pdf">An Easier Way to Show zeta(3) is Irrational</
               a>
%H A002117 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap97.html">Zeta(3) or Apery's 
               constant to 2000 places</a>
%H A002117 A. van der Poorten, <a href="http://www.ift.uni.wroc.pl/~mwolf/Poorten_MI_195_0.pdf">
               A Proof that Euler Missed</a>
%H A002117 Tanguy Rivoal, <a href="http://algo.inria.fr/seminars/sem01-02/rivoal.ps">
               Title?</a>
%H A002117 G. Villemin's Almanach of Numbers, <a href="http://perso.wanadoo.fr/yoda.guillaume/
               UnP2.htm#Apery">Apery's Constant(Text in French)</a>
%H A002117 S. Wedeniwski, <a href="http://www.ibiblio.org/gutenberg/etext01/zeta310.txt">
               The value of zeta(3) to 1000000 places</a> [Gutenberg Project Etext]
%H A002117 S. Wedeniwski, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               Zeta3.txt">Apery's constant to 128000026 decimal digits</a>
%H A002117 S. Wedeniwski, <a href="http://ftp.ibiblio.org/pub/docs/books/gutenberg/
               etext01/zeta310.txt">The value of zeta(3) to 1000000 decimal digits</
               a>
%H A002117 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AperysConstant.html">Link to a section of The World of Mathematics.</
               a>
%H A002117 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RelativelyPrime.html">'Relatively Prime'</a>
%H A002117 Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">
               Riemann zeta function</a>
%H A002117 H. Wilf, <a href="http://www.dmtcs.org/volumes/abstracts/dm030406.abs.html">
               Accelerated series for universal constants, by the WZ method</a>
%H A002117 Wadim Zudilin, <a href="http://arXiv.org/abs/math/0202159">An elementary 
               proof of Apery's theorem</a>
%H A002117 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>, Oct 
               14, 2009 [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 
               2009]
%F A002117 Lima conjectures that zeta(3) = (-5/197) + (11/394)*(pi^2)*(ln(2)) + 
               (283/394)*(pi)*(ln(2)^2) + (236/197)*(ln(3)^3) + (209/394)*(ln(1+sqrt(2)^3) 
               + (93*pi*gamma)/197 where gamma is the Euler-Mascheroni constant. 
               [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]
%e A002117 1.2020569031595942853997...
%t A002117 RealDigits[ N[ Zeta[3], 100] ] [ [1] ]
%o A002117 (PARI) { default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); 
               x=(x-d)*10; write("b002117.txt", n, " ", d)); } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Apr 19 2009]
%Y A002117 Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, 
               A013677.
%Y A002117 Cf. A059956 for 6/Pi^2.
%Y A002117 Cf. A084225; A084226.
%Y A002117 Sequence in context: A011420 A035686 A037228 this_sequence A042970 A158327 
               A136581
%Y A002117 Adjacent sequences: A002114 A002115 A002116 this_sequence A002118 A002119 
               A002120
%K A002117 cons,nonn,nice
%O A002117 1,2
%A A002117 N. J. A. Sloane (njas(AT)research.att.com).
%E A002117 More terms from David W. Wilson (davidwwilson(AT)comcast.net). Additional 
               comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 08 2000
%E A002117 Quotation from Stan Wagon corrected by N. J. A. Sloane (njas(AT)research.att.com) 
               on Dec 24 2005. Thanks to Jose Brox for noticing this error.
%E A002117 Fixed PARI Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 17 2009
%E A002117 New arXiv paper conjectures rational approximation for zeta(3). [From 
               Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]

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