A354953 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^5).
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, 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, 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
Offset: 1
Examples
3.359898760127253088364274368063313570407472689603469004194863140645872...
Links
- R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.
Programs
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Mathematica
digits = 105; precision = digits + 15; tmax = 400; (* integrand considered negligible beyond tmax *) kmax = 400; (* f(k) considered negligible beyond kmax *) InLogZeta[k_] := NIntegrate[(t-k)^4 Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision]; f[k_] := With[{mu = MoebiusMu[k]}, If[mu==0, 0, (mu/(4! k^6))* InLogZeta[k]]]; s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}]; RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 23 2022 *) -
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));
Extensions
Last 5 digits corrected by Vaclav Kotesovec, Jun 22 2022, following a suggestion from Jean-François Alcover