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%I #59 Feb 08 2024 07:13:05
%S 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,
%T 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,
%U 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
%N Decimal expansion of the prime zeta function at 3.
%C Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - _Jason Kimberley_, Jan 05 2017
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
%H Jason Kimberley, <a href="/A085541/b085541.txt">Table of n, a(n) for n = 0..1497</a>
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, Preprint, 1998.
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
%H Gerhard Niklasch and Pieter Moree, <a href="/A001692/a001692.html">Some number-theoretical constants</a> [Cached copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%F 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
%F Equals A086033 + A085992 + 1/8. - _R. J. Mathar_, Jul 22 2010
%F Equals Sum_{k>=1} 1/A030078(k). - _Amiram Eldar_, Jul 27 2020
%e 0.1747626392994435364231...
%t (* 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 *)
%o (PARI) recip3(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^3; ); print(v) }
%o (PARI) sumeulerrat(1/p,3) \\ _Hugo Pfoertner_, Feb 03 2020
%o (Magma) R := RealField(106);
%o PrimeZeta := func<k,N|
%o &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R,k*n)):n in[1..N]]>;
%o Reverse(IntegerToSequence(Floor(PrimeZeta(3,117)*10^105)));
%o // _Jason Kimberley_, Dec 30 2016
%Y Decimal expansion of the prime zeta function: A085548 (at 2), this sequence (at 3), A085964 (at 4) to A085969 (at 9).
%Y Cf. A002117, A030078, A242302.
%K easy,nonn,cons
%O 0,2
%A _Cino Hilliard_, Jul 02 2003
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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