login
Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.
8

%I #21 Jun 22 2022 09:35:50

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

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

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

%t digits = 106; precision = digits + 10;

%t tmax = 500; (* integrand considered negligible beyond tmax *)

%t kmax = 300; (* f(k) considered negligible beyond kmax *)

%t InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];

%t f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];

%t s = 0;

%t Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];

%t RealDigits[s][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Jun 21 2022, after _Vaclav Kotesovec_ *)

%o (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 Cf. A137245, A115563, A221711, A319231.

%K nonn,cons

%O 0,1

%A _R. J. Mathar_, Sep 14 2018

%E More terms from _Vaclav Kotesovec_, Jun 12 2022