A340879 Decimal expansion of K4 = 7/Pi^3 * Product_{primes p == 1 (mod 4)} (1 - 4/(p+1)^2).

1, 9, 0, 8, 7, 6, 7, 2, 1, 1, 6, 8, 5, 2, 8, 4, 4, 8, 0, 1, 1, 2, 2, 3, 7, 2, 4, 1, 3, 1, 1, 7, 1, 0, 8, 8, 3, 1, 4, 0, 9, 3, 4, 7, 9, 8, 3, 7, 0, 9, 6, 0, 4, 3, 3, 2, 8, 6, 7, 0, 2, 0, 4, 5, 8, 8, 6, 6, 2, 4, 6, 8, 5, 8, 5, 6, 5, 7, 7, 9, 1, 4, 2, 2, 8, 2, 8, 9, 4, 9, 5, 1, 2, 4, 4, 3, 2, 3, 8, 2, 9, 4, 4, 7, 2, 3

Offset: 0

Author

Vaclav Kotesovec, Jan 25 2021

Keywords

• nonn
• cons

Comments

The constant K4 from the paper by Finch and Sebah, p. 8. For more info see A340857.

Equal to the constant 2*C4 = (d(4) - 1)*C4 from the paper by Finch, Martin and Sebah, p. 2730, formula (5).

Examples

			0.190876721168528448011223724131171088314093479837096043328670204588662...
		

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. 8.

Crossrefs

Cf. A340857, A340878.

Programs

Mathematica

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := 1/(1 - 1/p^2)*(1 - (5 p - 3)/(p^2*(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[4, 1, m]; sump = sump + difp; m++];
RealDigits[Chop[N[7/Pi^3 * Exp[sump], digits]], 10, digits-1][[1]]

Formula

Equals 7/Pi^3 * Product_{primes p == 1 (mod 4)} 1/(1 - 1/p^2)*(1 - (5*p - 3)/(p^2*(p+1))).