cp's OEIS Frontend

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

%I A319232 #21 Jun 22 2022 09:35:50
%S A319232 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,
%T A319232 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,
%U A319232 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
%N A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.
%C A319232 Obtained by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.
%H A319232 R. J. Mathar, <a href="https://arxiv.org/abs/0811.4739">Twenty digits of some integrals of the prime zeta function</a>, arXiv:0811.4739 (2008-2009).
%e A319232 1/A016627^2 + 1/A016650^2 + 1/8.047189^2 + ... = 0.637056184074676....
%t A319232 digits = 106; precision = digits + 10;
%t A319232 tmax = 500; (* integrand considered negligible beyond tmax *)
%t A319232 kmax = 300; (* f(k) considered negligible beyond kmax *)
%t A319232 InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
%t A319232 f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
%t A319232 s = 0;
%t A319232 Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
%t A319232 RealDigits[s][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Jun 21 2022, after _Vaclav Kotesovec_ *)
%o A319232 (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
%Y A319232 Cf. A137245, A115563, A221711, A319231.
%K A319232 nonn,cons
%O A319232 0,1
%A A319232 _R. J. Mathar_, Sep 14 2018
%E A319232 More terms from _Vaclav Kotesovec_, Jun 12 2022