A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

6, 3, 7, 0, 5, 6, 1, 8, 4, 0, 7, 4, 6, 7, 6, 4, 3, 3, 0, 5, 9, 9, 6, 8, 5, 8, 5, 0, 4, 7, 8, 5, 2, 7, 6, 9, 4, 5, 7, 9, 8, 9, 6, 0, 7, 7, 1, 9, 9, 5, 3, 3, 6, 7, 0, 9, 6, 0, 1, 3, 7, 1, 0, 7, 5, 5, 8, 8, 3, 1, 6, 0, 4, 3, 3, 2, 7, 1, 5, 1, 6, 8, 3, 6, 7, 5, 3, 8, 3, 5, 9, 6, 6, 1, 3, 3, 1, 8, 1, 3, 1, 3, 8, 2, 7, 5

Offset: 0

Author

R. J. Mathar, Sep 14 2018

Keywords

• nonn
• cons

Comments

Obtained by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.

Examples

			1/A016627^2 + 1/A016650^2 + 1/8.047189^2 + ... = 0.637056184074676....
		

Links

• R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 (2008-2009).

Crossrefs

Cf. A137245, A115563, A221711, A319231.

Programs

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 *)

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

Extensions

More terms from Vaclav Kotesovec, Jun 12 2022