|
%I #21 Jun 22 2022 12:20:33
%S 2,0,0,6,6,6,6,4,5,2,8,3,1,0,6,8,7,5,6,4,3,2,2,9,6,9,9,9,4,7,1,3,5,8,
%T 2,0,8,4,8,8,6,8,3,5,4,1,4,7,5,0,4,5,7,8,0,5,9,0,5,4,9,8,2,7,8,2,7,4,
%U 7,8,2,1,9,2,1,6,4,7,0,5,5,0,3,1,8,4,3,8,1,7,5,9,2,0,1,5,6,1,0,1,3,0,7,9,6
%N Decimal expansion of the constant sum 1/(q*log(q)), summed over prime powers q > 1.
%C Evaluated from Sum_{m,k >= 1} A008683(k)*I(k*m)/k^2, where I(x) = Integral_{t=x..infinity} log zeta(t) dt is Cohen's underivative.
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H D. A. Clark, <a href="http://dx.doi.org/10.1016/S0898-1221(97)00293-9">An upper bound of sum 1/(a_i log a_i) for quasi-primitive sequences</a>, Comp. Math. Appl., 35 (1998), 105-109.
%H H. Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998.
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H R. J. Mathar, <a href="http://arxiv.org/abs/0811.4739">Twenty digits of some Integrals of the Prime Zeta Function</a>, arXiv:0811.4739 [math.NT].
%F Equals Sum_{n>=2} 1/(A000961(n)*log(A000961(n))).
%F Equals Sum_{p primes} -log(1-1/p)/log(p). - _Vaclav Kotesovec_, Jun 12 2022
%e 2.0066664528310687...
%o (PARI) default(realprecision, 200); su = 0; for(s=1, 400, su = su + sum(k=1, 500, moebius(k)/k^2 * intnum(x=s*k,[[1], 1], log(zeta(x))))/s; print(su)); \\ _Vaclav Kotesovec_, Jun 12 2022
%Y Cf. A137245, A221711.
%K nonn,cons
%O 1,1
%A _R. J. Mathar_, Mar 09 2008
%E 8 more digits from _R. J. Mathar_, Dec 04 2008
%E More terms from _Vaclav Kotesovec_, Jun 12 2022
|
