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A340878
Decimal expansion of K3 = 11*sqrt(3)/(18*Pi) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))).
2
3, 1, 7, 0, 5, 6, 5, 1, 6, 7, 9, 2, 2, 8, 4, 1, 2, 0, 5, 6, 7, 0, 1, 5, 6, 4, 0, 7, 1, 5, 0, 0, 6, 3, 6, 8, 1, 6, 7, 8, 5, 2, 6, 8, 7, 4, 8, 9, 1, 8, 4, 4, 2, 4, 3, 1, 4, 8, 4, 0, 9, 8, 7, 5, 9, 8, 7, 1, 8, 1, 5, 4, 4, 5, 9, 2, 4, 3, 2, 2, 6, 3, 8, 2, 1, 8, 8, 9, 3, 9, 8, 4, 9, 0, 1, 7, 1, 7, 7, 0, 9, 9, 1, 5, 1, 2
OFFSET
0,1
COMMENTS
The constant K3 from the paper by Finch and Sebah, p. 7. For more info see A340857.
Equal to the constant C3 = (d(3) - 1)*C3 from the paper by Finch, Martin and Sebah, p. 2730, formula (4).
LINKS
Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009, p. 7.
EXAMPLE
0.317056516792284120567015640715006368167852687489184424314840987598718...
MATHEMATICA
$MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 2/(p*(p + 1)));
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[3, 1, m]; sump = sump + difp; m++];
RealDigits[Chop[N[11*Sqrt[3]/(18*Pi)*Exp[sump], digits]], 10, digits-1][[1]]
CROSSREFS
Sequence in context: A128605 A051511 A272030 * A346182 A026499 A339452
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jan 25 2021
STATUS
approved