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

Author

R. J. Mathar, Mar 25 2012

Keywords

• nonn
• cons
• less

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

Examples

			0.88307100474394667141783429900310853467688883488097347...
		

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.

Crossrefs

Cf. A007519, A334826.

Programs

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 *)

Extensions

More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020

More digits from Vaclav Kotesovec, Jan 16 2021