A340577 - OEIS

%I #12 Aug 21 2022 09:43:32

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

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

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

%N Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2).

%F Equals 1.0004615.../A175646, both constants taken from p 27 of arXiv:1008.2537v2. - _R. J. Mathar_, Aug 21 2022

%e 0.96755040251956188660947077043906773001524912960304386356302398...

%t digits = 105;

%t precision = digits + 5;

%t prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];

%t Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;

%t Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];

%t Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];

%t gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];

%t pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;

%t pC = (2*Pi^4)/(243*pB*Lv[2]);

%t RealDigits[pC, 10, digits][[1]](* Most of this code is due to _Artur Jasinski_ *)

%t (* -------------------------------------------------------------------------- *)

%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 Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);

%t $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[6, 1, 4]/Z[6, 1, 2], digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)

%Y Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340578.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jan 12 2021

%E a(104) corrected by _Vaclav Kotesovec_, Jan 15 2021