OFFSET
0,1
COMMENTS
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
Named after the Austrian mathematician Emil Artin (1898-1962). - Amiram Eldar, Jun 20 2021
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Ivan Cherednik, A note on Artin's constant, arXiv:0810.2325 [math.NT], 2008.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2001; constant A_1^(1).
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Pieter Moree, The formal series Witt transform, Discr. Math., Vol. 295, No. 1-3 (2005), pp. 143-160. See p. 159.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
G. Niklasch, Artin's constant.
Simon Plouffe, The Artin's Constant=product(1-1/(p**2-p), p=prime) [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!]
Tomás Oliveira e Silva and Plouffe's Inverter, The first 500 digits of Artin's constant.
Eric Weisstein's World of Mathematics, Artin's constant.
Eric Weisstein's World of Mathematics, Full Reptend Prime.
R. G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
FORMULA
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
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
Equals 1/A065488. - Vaclav Kotesovec, Jul 17 2021
EXAMPLE
0.37395581361920228805472805434641641511162924860615...
MATHEMATICA
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 *)
PROG
(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
(PARI) prodeulerrat(1-1/(p^2-p)) \\ Amiram Eldar, Mar 12 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)
STATUS
approved