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A210630 Decimal expansion of Product_{primes p == 1 (mod 8)} p*(p-8)/(p-4)^2.

%I A210630 #27 Jan 16 2021 05:14:16
%S A210630 8,8,3,0,7,1,0,0,4,7,4,3,9,4,6,6,7,1,4,1,7,8,3,4,2,9,9,0,0,3,1,0,8,5,
%T A210630 3,4,6,7,6,8,8,8,8,3,4,8,8,0,9,7,3,4,7,0,7,1,9,2,9,5,1,5,9,3,9,5,2,1,
%U A210630 1,9,4,6,9,9,0,6,5,6,5,9,6,8,8,5,7,9,9,3,8,3,2,8,6,0,3,7,9,1,6,4,6,3,5,8,5,2
%N A210630 Decimal expansion of Product_{primes p == 1 (mod 8)} p*(p-8)/(p-4)^2.
%C A210630 Equals the product_{s>=2} of 1/zeta_(8,1)(s)^gamma(s), where gamma(s) = 16, 128, 888, 6144, 42256, 293912,... is an Euler transformation of the associated polynomial (1/x)(1/x-8)/(1/x-4)^2, and where the zeta_(m,n)(s) are the zeta prime modulo functions defined in section 3.3 of arXiv:1008.2547.
%C A210630 Note that Product_{k>=1} (8*k-7) * (8*k+1) / (8*k-3)^2 = Pi * 2^(9/2) * Gamma(1/4)^2 / Gamma(1/8)^4 = 0.290040073098462288674... - _Vaclav Kotesovec_, May 13 2020
%H A210630 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 (Corollary 1.9).
%H A210630 Daniel Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223315-8">Lal's constant and generalizations</a>, Math. Comp. 21 (100) (1967) 705-707.
%e A210630 0.88307100474394667141783429900310853467688883488097347...
%t A210630 $MaxExtraPrecision = 1000; digits = 121;
%t A210630 f[p_] := p*(p - 8)/(p - 4)^2;
%t A210630 coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
%t A210630 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 A210630 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 A210630 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 A210630 RealDigits[Chop[N[f[17]*Exp[sump], digits]], 10, digits - 1][[1]] (* _Vaclav Kotesovec_, Jan 16 2021 *)
%Y A210630 Cf. A007519, A334826.
%K A210630 nonn,cons,less
%O A210630 0,1
%A A210630 _R. J. Mathar_, Mar 25 2012
%E A210630 More digits from Ettahri article added by _Vaclav Kotesovec_, May 12 2020
%E A210630 More digits from _Vaclav Kotesovec_, Jan 16 2021