|
%I #18 Jun 23 2022 12:33:07
%S 3,3,5,9,8,9,8,7,6,0,1,2,7,2,5,3,0,8,8,3,6,4,2,7,4,3,6,8,0,6,3,3,1,3,
%T 5,7,0,4,0,7,4,7,2,6,8,9,6,0,3,4,6,9,0,0,4,1,9,4,8,6,3,1,4,0,6,4,5,8,
%U 7,2,3,3,6,8,8,3,0,4,0,4,7,7,9,2,1,0,9,8,5,4,8,4,1,4,3,9,2,3,5,5,8,0,8,2,0
%N Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^5).
%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 3.359898760127253088364274368063313570407472689603469004194863140645872...
%t digits = 105; precision = digits + 15;
%t tmax = 400; (* integrand considered negligible beyond tmax *)
%t kmax = 400; (* f(k) considered negligible beyond kmax *)
%t InLogZeta[k_] := NIntegrate[(t-k)^4 Log[Zeta[t]], {t, k, tmax},
%t WorkingPrecision -> precision, MaxRecursion -> 20,
%t AccuracyGoal -> precision];
%t f[k_] := With[{mu = MoebiusMu[k]}, If[mu==0, 0, (mu/(4! k^6))* InLogZeta[k]]];
%t s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
%t RealDigits[s][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Jun 23 2022 *)
%o (PARI) default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/(4!*k^6) * intnum(x=k,[[1], 1], (x-k)^4 * log(zeta(x))); print(s));
%Y Cf. A137245, A319231, A354917, A354954.
%K nonn,cons
%O 1,1
%A _Vaclav Kotesovec_, Jun 13 2022
%E Last 5 digits corrected by _Vaclav Kotesovec_, Jun 22 2022, following a suggestion from _Jean-François Alcover_
|
