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

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

Author

Eric W. Weisstein, Aug 22 2008

Keywords

• nonn
• cons

Comments

This constant appears in Franz Mertens's publication from 1874 on p. 58 (see link). - Artur Jasinski, Mar 17 2021

Examples

			-0.315718452053890076851... [corrected by _Georg Fischer_, Jul 29 2021]
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.2, p. 96.

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.

Crossrefs

Cf. A001620, A077761.

Programs

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 *)

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

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