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

0,2

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

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

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

Example

0.1747626392994435364231...

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 *)

Prog

(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
(Magma) R := RealField(106);
PrimeZeta := func<k, N|
&+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)):n in[1..N]]>;
Reverse(IntegerToSequence(Floor(PrimeZeta(3, 117)*10^105)));
// Jason Kimberley, Dec 30 2016

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.

Sequence in context: A343628 A085469 A050996 * A133055 A303133 A195384

Adjacent sequences: A085538 A085539 A085540 * A085542 A085543 A085544

Keyword

easy,nonn,cons

Author

Cino Hilliard, Jul 02 2003

Extensions

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

Status

approved