A137245 Decimal expansion of Sum_{p prime} 1/(p * log p).

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, 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, 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

Offset: 1

Author

R. J. Mathar, Mar 09 2008

Keywords

• cons
• nonn

Comments

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.

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]

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

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

Examples

			1.63661632335...
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.27.2, p. 186.

Links

• Karim Belabas and Henri Cohen, Computation of sum_{p prime} 1/(p^s log(p)), PARI/GP script, 2020.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), pp. 126-128, [DOI].
• P. Erdős, Some problems and results on combinatorial number theory, 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.
• Brady Haran and Jared Duker Lichtman, Primes and Primitive Sets, Numberphile video (2022).
• Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019.
• Jared Duker Lichtman, A proof of the Erdős primitive set conjecture, arXiv:2202.02384 [math.NT], 2022.
• Jared Duker Lichtman, Proving the Erdős Primitive Set Conjecture, Oxford Mathematics YouTube video, 2022.
• Jared Duker Lichtman, A proof of the Erdős primitive set conjecture, YouTube video from Combinatorial and additive number theory conference (CANT), 2022.
• R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2009, table in Section 2.4.
• David C. Ullrich, Re: What is Sum(1/p log p)?, posting in newsgroup sci.math.research, 11 Feb 2006.

Crossrefs

Cf. A000040, A060294, A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared), A354952.

Programs

Mathematica

digits = 105;
precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 06 2021, updated Jun 22 2022 *)

PARI

\\ See Belabas, Cohen link. Run as SumEulerlog(1) after setting the required precision.

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

Formula

Equals Sum_{n>=1} 1/(A000040(n)*log A000040(n)).

Extensions

More terms from Hugo Pfoertner, Feb 01 2020

More terms from Vaclav Kotesovec, Jun 12 2022