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 A143524 Decimal expansion of the (negated) constant in the expansion of the prime zeta function about s = 1. 3
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This constant appears in Franz Mertens's publication from 1874 on p. 58 (see link). - Artur Jasinski, Mar 17 2021 REFERENCES Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209. LINKS Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission] Carl-Erik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202. R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009, Table 2. Franz Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (1874), pp. 46-62 p. 58. Eric Weisstein's World of Mathematics, Prime Zeta Function Wikipedia, Prime zeta function. FORMULA Equals A077761 minus A001620. - R. J. Mathar, Jan 22 2009 From Amiram Eldar, Aug 08 2020: (Start) Equals -Sum{k>=2} mu(k) * log(zeta(k)) / k. Equals -Sum_{p prime} (1/p + log(1 - 1/p)) Equals Sum_{k>=2} P(k)/k, where P is the prime zeta function. (End) EXAMPLE -0.315718452053890076851... [corrected by Georg Fischer, Jul 29 2021] MATHEMATICA 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 *) CROSSREFS Cf. A001620, A077761. Sequence in context: A193844 A201552 A216182 * A134249 A188509 A265707 Adjacent sequences: A143521 A143522 A143523 * A143525 A143526 A143527 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Aug 22 2008 EXTENSIONS Digits changed to agree with A077761 and A001620 by R. J. Mathar, Oct 30 2009 Last digits corrected by Jean-François Alcover, Sep 11 2015 STATUS approved

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Last modified December 4 02:16 EST 2022. Contains 358544 sequences. (Running on oeis4.)