cp's OEIS Frontend

A143524 Decimal expansion of the (negated) constant in the expansion of the prime zeta function about s = 1.

%I A143524 #43 Feb 16 2025 08:33:08
%S A143524 3,1,5,7,1,8,4,5,2,0,5,3,8,9,0,0,7,6,8,5,1,0,8,5,2,5,1,4,7,3,7,0,6,5,
%T A143524 7,1,9,9,0,5,9,2,6,8,7,6,7,8,7,2,4,3,9,2,6,1,3,7,0,3,0,2,0,9,5,9,9,4,
%U A143524 3,2,1,5,8,8,0,2,9,6,4,6,1,2,2,2,8,0,4,4,3,1,8,5,7,5,0,0,0,9,8,4,6,3,0,1
%N A143524 Decimal expansion of the (negated) constant in the expansion of the prime zeta function about s = 1.
%C A143524 This constant appears in Franz Mertens's publication from 1874 on p. 58 (see link). - _Artur Jasinski_, Mar 17 2021
%D A143524 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D A143524 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.2, p. 96.
%H A143524 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High precision computation of Hardy-Littlewood constants</a>, preprint, 1998.
%H A143524 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A143524 Carl-Erik Fröberg, <a href="https://doi.org/10.1007/BF01933420">On the prime zeta function</a>, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
%H A143524 R. J. Mathar, <a href="http://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 [math.NT], 2008-2009, Table 2.
%H A143524 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/3325234/1039170">Prime Zeta function at 1</a>
%H A143524 Franz Mertens, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002155656">Ein Beitrag zur analytischen Zahlentheorie</a>, J. Reine Angew. Math. 78 (1874), pp. 46-62 p. 58.
%H A143524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.
%H A143524 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_zeta_function">Prime zeta function</a>.
%F A143524 Equals A077761 minus A001620. - _R. J. Mathar_, Jan 22 2009
%F A143524 From _Amiram Eldar_, Aug 08 2020: (Start)
%F A143524 Equals -Sum{k>=2} mu(k) * log(zeta(k)) / k.
%F A143524 Equals -Sum_{p prime} (1/p + log(1 - 1/p))
%F A143524 Equals Sum_{k>=2} P(k)/k, where P is the prime zeta function. (End)
%F A143524 P(s) = log(zeta(s)) - A143524 + o(1) = log(1/(s-1)) - A143524 + o(1) as s -> 1. - _Jianing Song_, Jan 10 2024
%e A143524 -0.315718452053890076851... [corrected by _Georg Fischer_, Jul 29 2021]
%t A143524 digits = 104; S = NSum[PrimeZetaP[n]/n, {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[S, 10, digits] // First (* _Jean-François Alcover_, Sep 11 2015 *)
%Y A143524 Cf. A001620, A077761.
%K A143524 nonn,cons
%O A143524 0,1
%A A143524 _Eric W. Weisstein_, Aug 22 2008
%E A143524 Digits changed to agree with A077761 and A001620 by _R. J. Mathar_, Oct 30 2009
%E A143524 Last digits corrected by _Jean-François Alcover_, Sep 11 2015