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%I #37 Jan 10 2024 11:02:41
%S 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 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 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 Decimal expansion of the (negated) constant in the expansion of the prime zeta function about s = 1.
%C This constant appears in Franz Mertens's publication from 1874 on p. 58 (see link). - _Artur Jasinski_, Mar 17 2021
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%H Henri 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 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 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 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/3325234/1039170">Prime Zeta function at 1</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_zeta_function">Prime zeta function</a>.
%F Equals A077761 minus A001620. - _R. J. Mathar_, Jan 22 2009
%F From _Amiram Eldar_, Aug 08 2020: (Start)
%F Equals -Sum{k>=2} mu(k) * log(zeta(k)) / k.
%F Equals -Sum_{p prime} (1/p + log(1 - 1/p))
%F Equals Sum_{k>=2} P(k)/k, where P is the prime zeta function. (End)
%F P(s) = log(zeta(s)) - A143524 + o(1) = log(1/(s-1)) - A143524 + o(1) as s -> 1. - _Jianing Song_, Jan 10 2024
%e -0.315718452053890076851... [corrected by _Georg Fischer_, Jul 29 2021]
%t 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 Cf. A001620, A077761.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Aug 22 2008
%E Digits changed to agree with A077761 and A001620 by _R. J. Mathar_, Oct 30 2009
%E Last digits corrected by _Jean-François Alcover_, Sep 11 2015
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