OFFSET
0,1
COMMENTS
Obtained by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.
LINKS
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 (2008-2009).
MATHEMATICA
digits = 106; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 300; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*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, 300, s = s + moebius(k)/k^3 * intnum(x=2*k, [[1], 1], (x-2*k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
R. J. Mathar, Sep 14 2018
EXTENSIONS
More terms from Vaclav Kotesovec, Jun 12 2022
STATUS
approved