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%I #13 Mar 16 2021 04:07:02
%S 1,4,7,3,4,9,4,0,0,0,0,2,0,0,1,4,5,8,0,7,6,8,0,8,4,3,1,8,4,7,6,9,2,5,
%T 9,6,3,9,6,6,7,1,8,5,8,1,7,3,2,7,2,1,5,8,4,4,2,0,7,9,6,1,9,2,8,5,5,5,
%U 8,3,5,3,4,0,9,8,5,5,0,3,5,5,9,8,0,7,8,2,7,1,1,3,0,1,7,6,6,1,8,9,9,4,4,3,3,6
%N Decimal expansion of Matthews' constant C_2, an analog of Artin's constant for primitive roots.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.
%H K. R. Matthews, <a href="https://eudml.org/doc/205412">A generalisation of Artin's conjecture for primitive roots</a>, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.
%F C_2 = Product_{p prime} 1 - (p^2 - (p - 1)^2)/(p^2*(p - 1)).
%F Log(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) + O(p,Infinity)^m = Sum_{n=2..m} -r(n)/(n*p^n), where r(n) = rootSum(1 - 2*x - x^2 + x^3, x^n) - 1.
%e 0.147349400002001458076808431847692596396671858173272158442...
%t digits = 66; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[1 - 2*# - #^2 + #^3& , #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m]][[1]]
%o (PARI) prodeulerrat(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) \\ _Amiram Eldar_, Mar 16 2021
%Y Cf. A005596.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Apr 14 2016
%E More digits from _Vaclav Kotesovec_, Jun 19 2020
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