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A143524
Decimal expansion of the (negated) constant in the expansion of the prime zeta function about s = 1.
10
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
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. [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.
Mathematics Stack Exchange, Prime Zeta function at 1
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
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)
P(s) = log(zeta(s)) - A143524 + o(1) = log(1/(s-1)) - A143524 + o(1) as s -> 1. - Jianing Song, Jan 10 2024
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
Sequence in context: A193844 A201552 A216182 * A134249 A188509 A265707
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