OFFSET
0,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.
LINKS
K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.
FORMULA
C_4 = Product_{p prime} 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)).
EXAMPLE
0.026107446314917708083...
MATHEMATICA
$MaxExtraPrecision = 2000; LR = LinearRecurrence[{2, 3, -10, 10, -5, 1}, {0, -8, 6, -40, 35, -194}, 10^4]; r[n_Integer] := LR[[n]]; NSum[r[n] PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2000, WorkingPrecision -> 300, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, 20]& // First // Prepend[#, 0]&
$MaxExtraPrecision = 1000; Clear[f]; f[p_] := 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)); Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 105]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)
PROG
(PARI) prodeulerrat(1 - (p^4 - (p - 1)^4)/(p^4*(p - 1))) \\ Amiram Eldar, Mar 16 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Apr 16 2016
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 19 2020
STATUS
approved