%I #12 Jan 25 2021 19:36:36
%S 1,9,0,8,7,6,7,2,1,1,6,8,5,2,8,4,4,8,0,1,1,2,2,3,7,2,4,1,3,1,1,7,1,0,
%T 8,8,3,1,4,0,9,3,4,7,9,8,3,7,0,9,6,0,4,3,3,2,8,6,7,0,2,0,4,5,8,8,6,6,
%U 2,4,6,8,5,8,5,6,5,7,7,9,1,4,2,2,8,2,8,9,4,9,5,1,2,4,4,3,2,3,8,2,9,4,4,7,2,3
%N Decimal expansion of K4 = 7/Pi^3 * Product_{primes p == 1 (mod 4)} (1 - 4/(p+1)^2).
%C The constant K4 from the paper by Finch and Sebah, p. 8. For more info see A340857.
%C Equal to the constant 2*C4 = (d(4) - 1)*C4 from the paper by Finch, Martin and Sebah, p. 2730, formula (5).
%H Steven Finch, Greg Martin and Pascal Sebah, <a href="http://www.math.ubc.ca/~gerg/papers/downloads/RUNM.pdf">Roots of unity and nullity modulo n</a>, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
%H Steven Finch and Pascal Sebah, <a href="https://arxiv.org/abs/0912.3677">Residue of a Mod 5 Euler Product</a>, arXiv:0912.3677 [math.NT], 2009, p. 8.
%F Equals 7/Pi^3 * Product_{primes p == 1 (mod 4)} 1/(1 - 1/p^2)*(1 - (5*p - 3)/(p^2*(p+1))).
%e 0.190876721168528448011223724131171088314093479837096043328670204588662...
%t $MaxExtraPrecision = 1000; digits = 121;
%t f[p_] := 1/(1 - 1/p^2)*(1 - (5 p - 3)/(p^2*(p + 1)));
%t coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
%t S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
%t P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]]*S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
%t m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[4, 1, m]; sump = sump + difp; m++];
%t RealDigits[Chop[N[7/Pi^3 * Exp[sump], digits]], 10, digits-1][[1]]
%Y Cf. A340857, A340878.
%K nonn,cons
%O 0,2
%A _Vaclav Kotesovec_, Jan 25 2021