%I #15 Jun 22 2022 09:25:46

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

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

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

%N Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^3).

%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 1.8461474193664495...

%t digits = 105; precision = digits + 15;

%t tmax = 500; (* integrand considered negligible beyond tmax *)

%t kmax = 500; (* f(k) considered negligible beyond kmax *)

%t InLogZeta[k_] := NIntegrate[(t - k)^2 Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];

%t f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/(2 k^4))*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)/(2*k^4) * intnum(x=k,[[1], 1], (x-k)^2 * log(zeta(x))); print(s));

%Y Cf. A137245, A145419, A221711, A319231, A319232.

%K nonn,cons

%O 1,2

%A _Vaclav Kotesovec_, Jun 12 2022

%E Last digit corrected by _Jean-François Alcover_ and confirmed by _Vaclav Kotesovec_, Jun 22 2022