A085968 Decimal expansion of the prime zeta function at 8.

0, 0, 4, 0, 6, 1, 4, 0, 5, 3, 6, 6, 5, 1, 7, 8, 3, 0, 5, 6, 0, 5, 2, 3, 4, 3, 9, 1, 4, 2, 6, 8, 3, 0, 8, 0, 5, 2, 2, 9, 7, 7, 1, 4, 4, 5, 1, 2, 0, 7, 1, 7, 4, 1, 0, 0, 1, 0, 3, 2, 6, 8, 8, 6, 8, 1, 7, 2, 8, 6, 3, 0, 4, 0, 7, 0, 7, 8, 8, 0, 4, 4, 0, 6, 0, 9, 2, 2, 8, 2, 8, 0, 5, 3, 0, 4, 3, 1, 3, 4, 4, 2, 6, 5, 6

Offset

0,3

Comments

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

References

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

Links

Jason Kimberley, Table of n, a(n) for n = 0..1971

Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

X. Gourdon and P. Sebah, Some Constants from Number theory

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.

Eric Weisstein's World of Mathematics, Prime Zeta Function

Formula

P(8) = Sum_{p prime} 1/p^8 = Sum_{n>=1} mobius(n)*log(zeta(8*n))/n.

Equals Sum_{k>=1} 1/A179645(k). - Amiram Eldar, Jul 27 2020

Example

0.0040614053665178305605...

Mathematica

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[8*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n = 200]; While[s[n] != s[n - 100], n = n + 100]; s[n] (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[ PrimeZetaP[ 8], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

Prog

(Magma) R := RealField(106);
PrimeZeta := func<k, N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)): n in[1..N]]>;
[0, 0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(8, 43)*10^105)));
// Jason Kimberley, Dec 30 2016
(PARI) sumeulerrat(1/p, 8) \\ Hugo Pfoertner, Feb 03 2020

Crossrefs

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085967 (at 7), this sequence (at 8), A085969 (at 9).

Cf. A013666, A179645.

Sequence in context: A132953 A195207 A157721 * A353005 A010637 A378333

Adjacent sequences: A085965 A085966 A085967 * A085969 A085970 A085971

Keyword

cons,easy,nonn

Author

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Status

approved