A085964 Decimal expansion of the prime zeta function at 4.

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

Offset: 0

Author

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

Keywords

• cons
• easy
• nonn

Comments

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

Examples

			0.0769931397642468449426...
		

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..1603
• 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

Crossrefs

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), this sequence (at 4), A085965 (at 5) to A085969 (at 9).

Cf. A013662, A030514, A242303.

Programs

Magma

R := RealField(106);
PrimeZeta := func;
[0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(4,87)*10^105)));
// Jason Kimberley, Dec 30 2016

Mathematica

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[4*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[ 4], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

PARI

sumeulerrat(1/p,4) \\ Hugo Pfoertner, Feb 03 2020

Formula

P(4) = Sum_{p prime} 1/p^4 = Sum_{n>=1} mobius(n)*log(zeta(4*n))/n

Equals A086034 + A085993 + 1/16. - R. J. Mathar, Jul 22 2010

Equals Sum_{k>=1} 1/A030514(k). - Amiram Eldar, Jul 27 2020