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A188596
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Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).
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1
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1, 5, 2, 1, 7, 3, 1, 5, 3, 5, 0, 7, 5, 7, 0, 5, 8, 1, 8, 8, 4, 1, 9, 5, 9, 4, 3, 4, 2, 9, 1, 3, 1, 1, 6, 9, 4, 0, 8, 0, 8, 0, 2, 7, 9, 5, 9, 4, 5, 4, 5, 0, 8, 6, 0, 5, 0, 8, 1, 5, 7, 9, 1, 8, 4, 5, 8, 1, 7, 3, 8, 5, 1, 3, 5, 6, 8, 2, 0, 3, 3, 0, 1, 0, 8, 1, 1, 4, 6, 5, 9, 5, 6, 5, 6, 4, 5, 4, 2, 7, 8, 7, 6, 4, 5
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OFFSET
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1,2
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COMMENTS
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This is the principal scale factor in an estimate of the number of primes p not exceeding N such that p^2+p+1 is also prime [Bateman-Horn].
A102283 in the definition plays the role of the Dirichlet character modulo 3.
After splitting the product into the three modulo-3 classes of primes, this constant turns out to be the product of four factors.
One factor as mentioned by Bateman and Horn is the inverse of A073010.
The second factor is 3/4 arising from the prime 3 which is the sole prime in the class == 0 (mod 3).
The third factor is product_{p == 1 (mod 3)} (1-(3p-1)/(p-1)^3) = 0.8675121817.. which is the constant C(m=3,n=1,s=3) of the arXiv preprint, basically the C(3) variant of A065418 reduced to the modulo class.
The final factor is product_{p == 2 (mod 3)} (1+1/(p^2-1)) = 1/product_{p == 2 (mod 3)} (1-1/p^2) = 1.41406439089214763.. which is the constant zeta(m=3,n=2,s=2) of the preprint and mentioned in A175646.
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LINKS
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EXAMPLE
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Equals 1.5217315350757058188419.. = 0.92003856361849186... / A073010 .
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MAPLE
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a073010 := evalf(Pi/3/sqrt(3)) ;
Cm3n0s2 := 1-1/(3-1)^2 ;
Cm3n1s3 := 0.867512181712394919089076584762888869720269526863 ;
Zm3n2s2 := 1.4140643908921476375655018190798293799076950693931 ;
Cm3n0s2*Cm3n1s3*Zm3n2s2/a073010 ;
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MATHEMATICA
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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}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[3^(5/2)*Zs[3, 1, 3]*Z[3, 2, 2]/(4*Pi), digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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