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%I #88 Jun 24 2022 05:29:25
%S 1,6,3,6,6,1,6,3,2,3,3,5,1,2,6,0,8,6,8,5,6,9,6,5,8,0,0,3,9,2,1,8,6,3,
%T 6,7,1,1,8,1,5,9,7,0,7,6,1,3,1,2,9,3,0,5,8,6,0,0,3,0,4,9,1,9,7,8,1,3,
%U 3,9,9,7,4,4,6,7,9,4,6,9,8,6,5,4,7,0,0,4,0,3,8,5,2,5,5,8,4,7,9,8,9,8,9,4,4
%N Decimal expansion of Sum_{p prime} 1/(p * log p).
%C Sum_{p prime} 1/(p^s * log p) equals this value here if s=1, equals A221711 if s=2, 0.22120334039... if s=3. See arXiv:0811.4739.
%C Erdős (1935) proved that for any sequence where no term divides another, the sum of 1/(x log x) is at most some constant C. He conjectures (1989) that C can be taken to be this constant 1.636..., that is, the primes maximize this sum. - _Charles R Greathouse IV_, Mar 26 2012 [The conjecture has been proved by Lichtman 2022. - _Pontus von Brömssen_, Jun 23 2022]
%C Note that sum 1/(p * log p) is almost (a tiny bit less than) 1 + 2/Pi = 1+A060294 = 1.63661977236758... (Why is it so close?) - _Daniel Forgues_, Mar 26 2012
%C Sum 1/(p * log p) is quite close to sum 1/n^2 = Pi^2/6 = 1.644934066... (Cf. David C. Ullrich, "Re: What is Sum(1/p log p)?" for why this is so; mentions A115563.) - _Daniel Forgues_, Aug 13 2012
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H 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 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision computation of Hardy-Littlewood constants</a>, (1998).
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1935-04.pdf">Note on sequences of integers no one of which is divisible by any other</a>, J. London Math. Soc. 10 (1935), pp. 126-128, <a href="http://dx.doi.org/10.1112/jlms/s1-10.1.126">[DOI]</a>.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1989-35.pdf">Some problems and results on combinatorial number theory</a>, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576, pp. 132-145, New York Acad. Sci., New York, 1989.
%H Brady Haran and Jared Duker Lichtman, <a href="https://www.numberphile.com/videos/primes-and-primitive-sets">Primes and Primitive Sets</a>, Numberphile video (2022).
%H Jared Duker Lichtman, <a href="https://arxiv.org/abs/1909.00804">Almost primes and the Banks-Martin conjecture</a>, arXiv:1909.00804 [math.NT], 2019.
%H Jared Duker Lichtman, <a href="https://arxiv.org/abs/2202.02384">A proof of the Erdős primitive set conjecture</a>, arXiv:2202.02384 [math.NT], 2022.
%H Jared Duker Lichtman, <a href="https://www.youtube.com/watch?v=c81ilGlIH1I">Proving the Erdős Primitive Set Conjecture</a>, Oxford Mathematics YouTube video, 2022.
%H Jared Duker Lichtman, <a href="https://www.youtube.com/watch?v=IsZ8Ae64zS0">A proof of the Erdős primitive set conjecture</a>, YouTube video from Combinatorial and additive number theory conference (CANT), 2022.
%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 [math.NT], 2008-2009, table in Section 2.4.
%H David C. Ullrich, <a href="https://groups.google.com/d/msg/sci.math.research/pp3fnEdVtok/Qb0g-2RJAGsJ">Re: What is Sum(1/p log p)?</a>, posting in newsgroup sci.math.research, 11 Feb 2006.
%F Equals Sum_{n>=1} 1/(A000040(n)*log A000040(n)).
%e 1.63661632335...
%t digits = 105;
%t precision = digits + 10;
%t tmax = 500; (* integrand considered negligible beyond tmax *)
%t kmax = 500; (* f(k) considered negligible beyond kmax *)
%t InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20];
%t f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*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_, Feb 06 2021, updated Jun 22 2022 *)
%o (PARI) See Belabas, Cohen link. Run as SumEulerlog(1) after setting the required precision.
%o (PARI) default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/k^2 * intnum(x=k,[[1], 1], log(zeta(x))); print(s)); \\ _Vaclav Kotesovec_, Jun 12 2022
%Y Cf. A000040, A060294, A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared), A354952.
%K cons,nonn
%O 1,2
%A _R. J. Mathar_, Mar 09 2008
%E More terms from _Hugo Pfoertner_, Feb 01 2020
%E More terms from _Vaclav Kotesovec_, Jun 12 2022
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