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A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).

%I A188596 #28 Nov 27 2024 07:04:39
%S A188596 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,
%T A188596 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,
%U A188596 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
%N A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).
%C A188596 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].
%C A188596 A102283 in the definition plays the role of the Dirichlet character modulo 3.
%C A188596 After splitting the product into the three modulo-3 classes of primes, this constant turns out to be the product of four factors.
%C A188596 One factor as mentioned by Bateman and Horn is the inverse of A073010.
%C A188596 The second factor is 3/4 arising from the prime 3 which is the sole prime in the class == 0 (mod 3).
%C A188596 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.
%C A188596 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.
%D A188596 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.
%H A188596 Paul T. Bateman and Roger A. Horn, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0148632-7">A heuristic asymptotic formula concerning the distribution of prime numbers</a>, Math. Comp. 16 (1962), 363-367, constant C.
%H A188596 H. Davenport and A. Schinzel, <a href="http://projecteuclid.org/euclid.ijm/1256055100">A note on certain arithmetical constants</a>, Illinois Math. J. 10 (2) (1966), 181-185.
%H A188596 R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%e A188596 Equals 1.5217315350757058188419... = 0.92003856361849186... / A073010 .
%p A188596 a073010 := evalf(Pi/3/sqrt(3)) ;
%p A188596 Cm3n0s2 := 1-1/(3-1)^2 ;
%p A188596 Cm3n1s3 := 0.867512181712394919089076584762888869720269526863 ;
%p A188596 Zm3n2s2 := 1.4140643908921476375655018190798293799076950693931 ;
%p A188596 Cm3n0s2*Cm3n1s3*Zm3n2s2/a073010 ;
%t A188596 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);
%t A188596 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}];
%t A188596 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]);
%t A188596 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]);
%t A188596 $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 *)
%Y A188596 Cf. A053182.
%K A188596 nonn,cons,less
%O A188596 1,2
%A A188596 _R. J. Mathar_, Apr 05 2011
%E A188596 More terms from _Vaclav Kotesovec_, Jan 16 2021