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.
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
Wikipedia, Prime zeta function.
FORMULA
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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Aug 22 2008
EXTENSIONS
Last digits corrected by Jean-François Alcover, Sep 11 2015
STATUS
approved