A340878 Decimal expansion of K3 = 11*sqrt(3)/(18*Pi) * Product_{primes p == 1 (mod 3)} (1 - 2/(p*(p+1))).

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

Table of n, a(n) for n=0..105.

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

Cf. A340857, A340879.

Sequence in context: A128605 A051511 A272030 * A346182 A026499 A339452

Adjacent sequences: A340875 A340876 A340877 * A340879 A340880 A340881

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jan 25 2021

Status

approved