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A210630
Decimal expansion of Product_{primes p == 1 (mod 8)} p*(p-8)/(p-4)^2.
2
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, 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, 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
OFFSET
0,1
COMMENTS
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.
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
LINKS
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.9).
Daniel Shanks, Lal's constant and generalizations, Math. Comp. 21 (100) (1967) 705-707.
EXAMPLE
0.88307100474394667141783429900310853467688883488097347...
MATHEMATICA
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := p*(p - 8)/(p - 4)^2;
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
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);
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}];
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++];
RealDigits[Chop[N[f[17]*Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
CROSSREFS
Sequence in context: A174235 A372776 A185266 * A377008 A265292 A272415
KEYWORD
nonn,cons,less
AUTHOR
R. J. Mathar, Mar 25 2012
EXTENSIONS
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jan 16 2021
STATUS
approved