A334826 Decimal expansion of Product_{primes p==1 mod 8} (1 - 4/p)*((p + 1)/(p - 1))^2.

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, 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, 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

Offset: 0

Author

Vaclav Kotesovec, May 13 2020

Keywords

• nonn
• cons

Comments

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...

Examples

			0.9569453478516011834369670572738918287531749772913914789...
		

Links

• Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
• Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Crossrefs

Cf. A007519, A210630.

Programs

Mathematica

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}];
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]);
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]);
$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 *)
(* -------------------------------------------------------------------------- *)
(* second program, more general *)
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (1 - 4/p)*((p + 1)/(p - 1))^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 *)