OFFSET
1,1
LINKS
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.
EXAMPLE
3.359898760127253088364274368063313570407472689603469004194863140645872...
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 *)
PROG
(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));
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Jun 13 2022
EXTENSIONS
Last 5 digits corrected by Vaclav Kotesovec, Jun 22 2022, following a suggestion from Jean-François Alcover
STATUS
approved