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%I #30 Mar 16 2021 04:06:58
%S 2,7,7,8,7,6,8,8,2,0,7,3,2,3,1,9,6,1,9,3,2,3,1,0,8,6,6,7,0,3,2,5,3,4,
%T 2,0,3,6,0,2,0,6,2,9,4,1,4,7,3,6,8,2,9,8,8,2,4,5,2,7,0,5,3,3,6,7,7,1,
%U 6,4,9,8,0,0,8,2,8,3,5,0,7,5,9,9,6,6,3,7,4,8,8,4,6,9,1,0,3,9,4,1,6,6,9,8,0,9,2,9,5,8,6,6,1
%N Log of twice the twin prime constant, C_2, log(2*A005597).
%C The value occurs as term in equation (15) in the Wolf paper. - _Ralf Stephan_, Mar 28 2014
%H Marek Wolf, <a href="https://doi.org/10.1103/PhysRevE.89.022922">Nearest-neighbor-spacing distribution of prime numbers and quantum chaos</a>, Phys. Rev. E 89, 022922 (2014); <a href="http://arxiv.org/abs/1212.3841">arXiv preprint</a>, arXiv:1212.3841 [math.NT], 2012-2014.
%F Equals log(2*A005597).
%e 0.2778768820732319619323108667032534203602062941473682988245270533677164980...
%t digits = 113;
%t s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 50];
%t C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 50}];
%t RealDigits[Log[2 C2]][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Feb 16 2019 *)
%o (PARI)
%o default(realprecision,1000);
%o result={175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))};log(2*result)
%o (PARI) log(2 * prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ _Amiram Eldar_, Mar 16 2021
%Y Cf. A005597, A114907.
%K nonn,cons,less
%O 0,1
%A _John W. Nicholson_, Mar 03 2014
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