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%I #30 Jun 23 2022 12:32:43
%S 1,5,2,0,9,7,0,4,3,9,9,3,9,5,0,0,8,6,3,4,6,1,4,2,8,6,2,8,6,1,5,5,7,9,
%T 5,2,1,9,5,6,8,4,6,1,6,7,7,6,8,3,5,0,1,1,0,6,5,5,5,2,7,5,3,5,9,6,3,4,
%U 1,0,6,4,4,3,1,0,4,1,0,4,7,2,0,6,6,3,0,7,6,1,9,5,2,2,5,2,7,5,1,3,3,4,4,6,0
%N Decimal expansion of Sum_{p = prime} 1/(p*log(p)^2).
%C Computed by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.
%H R. J. Mathar, <a href="https://arxiv.org/abs/0811.4739">Twenty digits of some integrals of the prime zeta function</a>, arXiv:0811.4739 [math.NT], 2008-2018.
%e 1/(2*A253191) + 1/(3*A175478) +1/(5*2.59029...) +1/(7*3.7865)+ ... = 1.52097043...
%t digits = 105; precision = digits + 10;
%t tmax = 500; (* integrand considered negligible beyond tmax *)
%t kmax = 500; (* f(k) considered negligible beyond kmax *)
%t InLogZeta[k_] := NIntegrate[(t - k) Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
%t f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
%t s = 0;
%t Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
%t RealDigits[s][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Jun 21 2022, after _Vaclav Kotesovec_ *)
%o (PARI) default(realprecision, 200); s=0; for(k=1, 500, s=s+moebius(k)/k^3 * intnum(x=k,[[1], 1],(x-k)*log(zeta(x))); print(s)); \\ _Vaclav Kotesovec_, Jun 12 2022
%Y Cf. A137245, A115563, A221711, A319232, A354917.
%K nonn,cons
%O 1,2
%A _R. J. Mathar_, Sep 14 2018
%E More digits from _Vaclav Kotesovec_, Jun 12 2022
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