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A221711 Decimal expansion of sum 1/(p^2 * log p) over the primes p=2,3,5,7,11,...

%I A221711 #33 Sep 09 2024 13:43:17
%S A221711 5,0,7,7,8,2,1,8,7,8,5,9,1,9,9,3,1,8,7,7,4,3,7,5,1,0,3,7,9,4,7,0,5,5,
%T A221711 7,0,4,6,6,9,7,3,6,7,1,7,0,4,3,2,0,6,9,8,5,7,3,9,8,0,2,1,2,3,4,8,2,7,
%U A221711 2,8,6,9,0,1,3,7,4,1,3,1,1,5,1,0,4,6,4,6,6,7,8,4,8,9,5,2,9,2,1,1,3,5,6,4,5,4
%N A221711 Decimal expansion of sum 1/(p^2 * log p) over the primes p=2,3,5,7,11,...
%D A221711 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H A221711 Karim Belabas and Henri Cohen, <a href="/A137245/a137245.gp.txt">Computation of sum_{p prime} 1/(p^s log(p))</a>, PARI/GP script, 2020.
%H A221711 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision calculation of Hardy-Littlewood constants</a>, (1998).
%H A221711 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A221711 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 [math.NT], 2009-2018, Section 2.4.
%e A221711 0.50778218785919931877437510379470557...
%t A221711 digits = 106; precision = digits + 15;
%t A221711 tmax = 400; (* integrand considered negligible beyond tmax *)
%t A221711 kmax = 400; (* f(k) considered negligible beyond kmax *)
%t A221711 InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax},
%t A221711    WorkingPrecision -> precision, MaxRecursion -> 20,
%t A221711    AccuracyGoal -> precision];
%t A221711 f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[2k]]];
%t A221711 s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
%t A221711 RealDigits[s][[1]][[1 ;; digits]] (* _Jean-François Alcover_, Feb 06 2021, updated Jun 23 2022 *)
%o A221711 (PARI) \\ See Belabas, Cohen link. Run as SumEulerlog(2) after setting the required precision.
%o A221711 (PARI) default(realprecision, 200); s=0; for(k=1, 300, s = s + moebius(k)/k^2 * intnum(x=2*k,[[1], 1], log(zeta(x))); print(s)); \\ _Vaclav Kotesovec_, Jun 12 2022
%Y A221711 Cf. A137245.
%K A221711 nonn,cons
%O A221711 0,1
%A A221711 _N. J. A. Sloane_, Jan 26 2013
%E A221711 More terms from _Hugo Pfoertner_, Feb 01 2020
%E A221711 More digits from _Vaclav Kotesovec_, Jun 12 2022