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%I #40 Jun 29 2023 09:03:14
%S 1,3,7,3,1,0,6,5,1,8,0,9,0,7,3,5,9,1,8,7,1,5,8,7,4,7,0,6,1,2,4,3,5,0,
%T 1,2,3,1,9,8,5,4,4,7,2,2,1,4,5,1,6,1,5,4,3,9,9,3,9,4,4,4,4,1,5,0,4,5,
%U 6,8,1,9,6,2,8,9,6,0,8,2,7,5,7,5,4,5,6
%N Decimal expansion of the probability that an integer 4-tuple is pairwise unitary coprime.
%C Two numbers are unitary coprime if their largest common unitary divisor is 1.
%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.
%H László Tóth, <a href="https://doi.org/10.1007/978-1-4939-1106-6_19">Multiplicative arithmetic functions of several variables: a survey</a>, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 509), <a href="https://arxiv.org/abs/1310.7053">preprint</a>, arXiv:1310.7053 [math.NT], 2013-2014 (see p. 22).
%F Equals zeta(2)^2 * zeta(3) * zeta(4) * Product_{p prime} (1 - 8/p^2 + 3/p^3 + 27/p^4 - 24/p^5 - 14/p^6 - 3/p^7 + 37/p^8 - 30/p^9 + 42/p^10 - 33/p^11 - 41/p^12 + 78/p^13 - 44/p^14 + 9/p^15).
%e 0.137310651809073591871587470612435012319854472214516...
%t $MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 8*x^2 + 3*x^3 + 27*x^4 - 24*x^5 - 14*x^6 - 3*x^7 + 37*x^8 - 30*x^9 + 42*x^10 - 33*x^11 - 41*x^12 + 78*x^13 - 44*x^14 + 9*x^15; c = LinearRecurrence[{-3, 2, 11, -3, -16, -14, 6, 7, 19, 0, -17, 9}, {0, -16, 9, -20, 0, 161, -588, 2116, -5859, 15104, -34716, 70609}, nm]; RealDigits[Zeta[2]^2*Zeta[3]*Zeta[4]*f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]
%o (PARI) zeta(2)^2 * zeta(3) * zeta(4) * prodeulerrat(1-8/p^2+3/p^3+27/p^4-24/p^5-14/p^6-3/p^7+37/p^8-30/p^9+42/p^10-33/p^11-41/p^12+78/p^13-44/p^14+9/p^15) \\ _Amiram Eldar_, Jun 29 2023
%Y Cf. A077610, A306071, A319592.
%K nonn,cons
%O 0,2
%A _Amiram Eldar_, Aug 27 2019
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