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%I #8 Feb 11 2022 10:39:42
%S 8,5,9,3,9,2,2,3,1,3,5,8,5,6,8,6,8,9,7,1,8,7,1,4,5,1,4,1,8,6,1,2,3,2,
%T 8,1,7,6,9,9,6,0,9,1,7,6,9,8,3,1,1,2,1,1,4,7,4,1,6,3,4,2,6,5,9,0,3,8,
%U 3,9,6,4,9,4,1,6,7,1,1,1,3,1,3,6,3,1,7,2,1,4,3,9,6,2,2,2,8,6,5,8,3,8,0,6,6,6
%N Decimal expansion of Sum_{primes p > 2} log(p) / ((p-2)*(p-1)).
%C Constant is related to the asymptotics of A069205.
%H Emil Grosswald, <a href="http://doi.org/10.1215/S0012-7094-56-02305-5">The average order of an arithmetic function</a>, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44. [constant C3]
%e 0.8593922313585686897187145141861232817699609176983112114741634265903839649...
%t ratfun = 1/((p-2)*(p-1)); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(Zeta'[power]/Zeta[power] + Log[2]/(2^power - 1)) /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 2, m}] + zetas, 110]], {m, 2000, 10000, 2000}]
%Y Cf. A069205, A351347.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Aug 22 2021
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