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Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).
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%I #16 Sep 18 2022 02:07:44

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

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

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

%N Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

%C Also, the asymptotic mean of A162511. - _Amiram Eldar_, Sep 18 2022

%H V. Sitaramaiah and M. V. Subbarao, <a href="https://doi.org/10.1007/BFb0086405">Some asymptotic formulae involving powers of arithmetic functions</a>, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, <a href="https://www.researchgate.net/profile/V_Sitaramaiah/publication/227069035_Some_asymptotic_formulae_involving_powers_of_arithmetic_functions/links/5621cd0d08aed8dd1943e975.pdf">alternative link</a>.

%F Equals lim_{m->oo} (1/m)*Sum_{k=1..m} phi(k)/psi(k).

%F Equals Product_{p prime} (1 - 2/(p * (p+1))).

%F Equals A065472 / zeta(2). - _Amiram Eldar_, Sep 18 2022

%e 0.47168061361299786807523563308048208742592638200698...

%t $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 1, 2}, {0, -4, 6}, m]; RealDigits[(2/3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

%o (PARI) prodeulerrat(1 - 2/(p*(p+1))) \\ _Vaclav Kotesovec_, Sep 19 2020

%Y Cf. A000010, A001615, A013661, A065472, A162511, A173557.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, May 02 2019

%E More digits from _Vaclav Kotesovec_, Sep 19 2020