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A340577 Decimal expansion of Product_{primes p == 1 (mod 6)} 1/(1+1/p^2). 11
9, 6, 7, 5, 5, 0, 4, 0, 2, 5, 1, 9, 5, 6, 1, 8, 8, 6, 6, 0, 9, 4, 7, 0, 7, 7, 0, 4, 3, 9, 0, 6, 7, 7, 3, 0, 0, 1, 5, 2, 4, 9, 1, 2, 9, 6, 0, 3, 0, 4, 3, 8, 6, 3, 5, 6, 3, 0, 2, 3, 9, 8, 0, 8, 4, 0, 6, 8, 7, 3, 9, 5, 1, 6, 3, 8, 3, 9, 9, 9, 4, 6, 1, 6, 0, 5, 4, 1, 7, 8, 7, 3, 7, 7, 4, 2, 2, 3, 6, 8, 7, 5, 9, 8, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..104.

EXAMPLE

0.96755040251956188660947077043906773001524912960304386356302398...

MATHEMATICA

digits = 105;

precision = digits + 5;

prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];

Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;

Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];

Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];

gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];

pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;

pC = (2*Pi^4)/(243*pB*Lv[2]);

RealDigits[pC, 10, digits][[1]](* Most of this code is due to Artur Jasinski *)

(* -------------------------------------------------------------------------- *)

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]);

$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[6, 1, 4]/Z[6, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

CROSSREFS

Similar constants: A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340578.

Sequence in context: A155129 A254980 A318336 * A229923 A203080 A197375

Adjacent sequences:  A340574 A340575 A340576 * A340578 A340579 A340581

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Jan 12 2021

EXTENSIONS

a(104) corrected by Vaclav Kotesovec, Jan 15 2021

STATUS

approved

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Last modified July 26 13:40 EDT 2021. Contains 346294 sequences. (Running on oeis4.)