OFFSET
1,2
LINKS
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.
EXAMPLE
1.8461474193664495...
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 *)
PROG
(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));
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jun 12 2022
EXTENSIONS
Last digit corrected by Jean-François Alcover and confirmed by Vaclav Kotesovec, Jun 22 2022
STATUS
approved