

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
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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. 208209.


LINKS

Table of n, a(n) for n=0..103.
Henri Cohen, High precision computation of HardyLittlewood constants, preprint, 1998.
Henri Cohen, Highprecision computation of HardyLittlewood constants. [pdf copy, with permission]
CarlErik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187202.
R. J. Mathar, Series of reciprocal powers of kalmost primes, arXiv:0803.0900 [math.NT], 20082009, Table 2.
Franz Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (1874), pp. 4662 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 (* JeanFrançois 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 JeanFrançois Alcover, Sep 11 2015


STATUS

approved



