%I #30 Mar 11 2021 10:48:27

%S 5,3,5,1,0,7,0,1,2,6,1,6,6,3,8,7,3,3,2,8,3,9,5,8,6,5,1,8,6,0,6,3,2,1,

%T 5,9,8,5,8,6,3,9,3,3,9,1,0,2,8,0,1,3,4,9,2,6,6,5,2,7,2,8,8,4,8,8,9,8,

%U 2,4,3,8,8,2,1,0,0,2,6,9,0,3,5,6,1,4,4,2,0,9,2,5,2,1,5,9,4,6,2

%N Decimal expansion of Product_{p odd prime} 1-2/(p*(p-1)), a constant related to Artin's conjecture in the context of quadratic fields.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

%F Equals (8/Pi^2)*A005597.

%e 0.5351070126166387332839586518606321598586393391028...

%t digits = 99; $MaxExtraPrecision = 600; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, 1, -2}, {0, 4, 6}, 2 m0]; r[n_Integer] := LR[[n]];

%t s[m_] := s[m] = NSum[-r[n] (PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 600] // 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], 10, digits][[1]] (* _Jean-François Alcover_, Apr 15 2016 *)

%o (PARI) prodeulerrat(1-2/(p*(p-1)), 1, 3) \\ _Amiram Eldar_, Mar 11 2021

%Y Cf. A005597, A217739 (8/Pi^2).

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Apr 15 2016