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

0,1

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

Table of n, a(n) for n=0..103.

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

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

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