A354917 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^3).

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, 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, 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

Offset: 1

Author

Vaclav Kotesovec, Jun 12 2022

Keywords

• nonn
• cons

Examples

			1.8461474193664495...
		

Links

• R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.

Crossrefs

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

Programs

Mathematica

digits = 105; precision = digits + 15;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - k)^2 Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/(2 k^4))*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 21 2022, after Vaclav Kotesovec *)

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));

Extensions

Last digit corrected by Jean-François Alcover and confirmed by Vaclav Kotesovec, Jun 22 2022