%I #14 Jan 25 2021 19:36:18

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

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

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

%N Decimal expansion of K3 = 11*sqrt(3)/(18*Pi) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))).

%C The constant K3 from the paper by Finch and Sebah, p. 7. For more info see A340857.

%C Equal to the constant C3 = (d(3) - 1)*C3 from the paper by Finch, Martin and Sebah, p. 2730, formula (4).

%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. 7.

%e 0.317056516792284120567015640715006368167852687489184424314840987598718...

%t $MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 2/(p*(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[3, 1, m]; sump = sump + difp; m++];

%t RealDigits[Chop[N[11*Sqrt[3]/(18*Pi)*Exp[sump], digits]], 10, digits-1][[1]]

%Y Cf. A340857, A340879.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Jan 25 2021