OFFSET
0,2
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).
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.
FORMULA
Equals 7/Pi^3 * Product_{primes p == 1 (mod 4)} 1/(1 - 1/p^2)*(1 - (5*p - 3)/(p^2*(p+1))).
EXAMPLE
0.190876721168528448011223724131171088314093479837096043328670204588662...
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jan 25 2021
STATUS
approved