OFFSET
1,2
LINKS
Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]
Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.
EXAMPLE
1.64933768909803...
MATHEMATICA
kmax = 25; Do[ P[k] = Product[p = Prime[n]; If[Mod[p, 4] == 3, 1 - 2/(p*(p - 1)^2) // N[#, 40]&, 1], {n, 1, 2^k}]; Print["P(", k, ") = ", P[k]], {k, 10, kmax}]; c = 2*P[kmax]; RealDigits[c, 10, 15] // First
(* -------------------------------------------------------------------------- *)
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (1 - 2/(p*(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[4, 3, m]; sump = sump + difp; m++];
RealDigits[Chop[N[2*Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Oct 17 2014
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 27 2020
STATUS
approved