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%I #15 Jan 16 2021 05:47:00
%S 9,5,6,9,4,5,3,4,7,8,5,1,6,0,1,1,8,3,4,3,6,9,6,7,0,5,7,2,7,3,8,9,1,8,
%T 2,8,7,5,3,1,7,4,9,7,7,2,9,1,3,9,1,4,7,8,9,0,5,4,3,2,6,0,4,2,4,6,0,1,
%U 7,0,1,6,4,4,4,8,8,8,8,5,9,4,8,1,4,4,0,5,1,2,0,3,9,0,7,9,5
%N Decimal expansion of Product_{primes p==1 mod 8} (1 - 4/p)*((p + 1)/(p - 1))^2.
%C Note that Product_{k>=1} (4*k + 1)^2 * (8*k - 3) / (16 * k^2 * (8*k + 1)) = 2^(1/4) * Gamma(1/8)^2 / (sqrt(Pi) * Gamma(1/4)^3) = 0.79906817873784592665354...
%H Salma Ettahri, Olivier Ramaré, Léon Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arXiv:1908.06808 [math.NT], 2019.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%e 0.9569453478516011834369670572738918287531749772913914789...
%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 Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
%t $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Zs[8, 1, 4]/Z[8, 1, 2]^2, digits]], 10, digits-1][[1]] (* _Vaclav Kotesovec_, Jan 15 2021 *)
%t (* -------------------------------------------------------------------------- *)
%t (* second program, more general *)
%t $MaxExtraPrecision = 1000; digits = 121;
%t f[p_] := (1 - 4/p)*((p + 1)/(p - 1))^2;
%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[8, 1, m] - 1/17^m); sump = sump + difp; m++];
%t RealDigits[Chop[N[f[17]*Exp[sump], digits]], 10, digits - 1][[1]] (* _Vaclav Kotesovec_, Jan 16 2021 *)
%Y Cf. A007519, A210630.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, May 13 2020
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