A085541 Decimal expansion of the prime zeta function at 3.

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

Offset: 0

Author

Cino Hilliard, Jul 02 2003

Keywords

• easy
• nonn
• cons

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

			0.1747626392994435364231...
		

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..1497
• 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.
• Gerhard Niklasch and Pieter Moree, Some number-theoretical constants [Cached copy]
• Eric Weisstein's World of Mathematics, Prime Zeta Function
• Index to constants which are prime zeta sums {3,0,0}

Crossrefs

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

Cf. A002117, A030078, A242302.

Programs

Magma

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(3,117)*10^105)));
// Jason Kimberley, Dec 30 2016

Mathematica

(* If Mathematica version >= 7.0 then RealDigits[PrimeZetaP[3]//N[#,105]&][[1]] else : *) m = 200; $MaxExtraPrecision = 200; PrimeZetaP[s_] := NSum[MoebiusMu[k]*Log[Zeta[k*s]]/k, {k, 1, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]; RealDigits[PrimeZetaP[3]][[1]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011 *)

PARI

recip3(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^3; ); print(v) }

PARI

sumeulerrat(1/p,3) \\ Hugo Pfoertner, Feb 03 2020

Formula

P(3) = Sum_{p prime} 1/p^3 = Sum_{n>=1} mobius(n)*log(zeta(3*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Equals A086033 + A085992 + 1/8. - R. J. Mathar, Jul 22 2010

Equals Sum_{k>=1} 1/A030078(k). - Amiram Eldar, Jul 27 2020

Extensions

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