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%I M2608 #103 Jul 17 2021 05:47:49
%S 3,7,3,9,5,5,8,1,3,6,1,9,2,0,2,2,8,8,0,5,4,7,2,8,0,5,4,3,4,6,4,1,6,4,
%T 1,5,1,1,1,6,2,9,2,4,8,6,0,6,1,5,0,0,4,2,0,9,4,7,4,2,8,0,2,4,1,7,3,5,
%U 0,1,8,2,0,4,0,0,2,8,0,8,2,3,4,4,3,0,4,3,1,7,0,8,7,2,5,0,5,6,8,9,8,1,6,0,3
%N Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).
%C On Simon Plouffe's web page (and in the book freely available at Gutenberg project) the value is given with an error of +1e-31, as "...651641..." instead of "...641641...". In the reference [Wrench, 1961] cited there, these digits are correct. They are also correct on the Plouffe's Inverter page, as computed by Oliveira e Silva, who comments it took 1 hour at 200 MHz with Mathematica. Using _Amiram Eldar_'s PARI program, the same 500 digits are computed instantly (less than 0.1 sec). - _M. F. Hasler_, Apr 20 2021
%C Named after the Austrian mathematician Emil Artin (1898-1962). - _Amiram Eldar_, Jun 20 2021
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harry J. Smith, <a href="/A005596/b005596.txt">Table of n, a(n) for n = 0..1000</a>
%H Ivan Cherednik, <a href="http://arxiv.org/abs/0810.2325">A note on Artin's constant</a>, arXiv:0810.2325 [math.NT], 2008.
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision computation of Hardy-Littlewood constants</a>, (1998).
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 [math.NT], 2009-2001; constant A_1^(1).
%H Pieter Moree, <a href="https://arxiv.org/abs/math/0412262">Artin's primitive root conjecture - a survey</a>, arXiv:math/0412262 [math.NT], 2004-2012.
%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math., Vol. 295, No. 1-3 (2005), pp. 143-160. See p. 159.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H G. Niklasch, <a href="https://web.archive.org/web/20001209004100/http://www.gn-50uma.de/alula/essays/Moree/Moree-details.en.shtml">Artin's constant</a>.
%H Simon Plouffe, <a href="https://web.archive.org/web/20030816184018/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap8.html">The Artin's Constant=product(1-1/p**2-p), p=prime)</a> [backup on web.archive.org; chapter 8 of the free Gutenberg.org/ebooks/634]. [Warning: the value given in this reference is incorrect, cf. comment!]
%H Tomás Oliveira e Silva and Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/artin.txt">The first 500 digits of Artin's constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FullReptendPrime.html">Full Reptend Prime</a>.
%H R. G. Wilson v, <a href="/A005596/a005596.pdf">Letter to N. J. A. Sloane, Aug. 1993</a>.
%H John W. Wrench, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1961-0124305-0">Evaluation of Artin's constant and the twin-prime constant</a>, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
%H <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>.
%F Equals Product_{j>=2} 1/Zeta(j)^A006206(j), where Zeta = A013661, A002117 etc. is Riemann's zeta function. - _R. J. Mathar_, Feb 14 2009
%F Equals Sum_{k>=1} mu(k)/(k*phi(k)), where mu is the Moebius function (A008683) and phi is the Euler totient function (A000010). - _Amiram Eldar_, Mar 11 2020
%F Equals 1/A065488. - _Vaclav Kotesovec_, Jul 17 2021
%e 0.37395581361920228805472805434641641511162924860615...
%t a = Exp[-NSum[ (LucasL[n] - 1)/n PrimeZetaP[n], {n, 2, Infinity}, PrecisionGoal -> 500, WorkingPrecision -> 500, NSumTerms -> 100000]]; RealDigits[a, 10, 111][[1]] (* _Robert G. Wilson v_, Sep 03 2014 taken from Mathematica's Help file on PrimeZetaP *)
%o (PARI) prodinf(n=2,1/zeta(n)^(sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)) \\ _Charles R Greathouse IV_, Aug 27 2014
%o (PARI) prodeulerrat(1-1/(p^2-p)) \\ _Amiram Eldar_, Mar 12 2021
%Y Cf. A048296, A065414, A001913, A001122.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_
%E More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)
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