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A085541
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Decimal expansion of the prime zeta function at 3.
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46
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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
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OFFSET
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0,2
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COMMENTS
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Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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0.1747626392994435364231...
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) recip3(n) = { v=0; p=1; forprime(y=2, n, v=v+1./y^3; ); print(v) }
(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)));
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CROSSREFS
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Decimal expansion of the prime zeta function: A085548 (at 2), this sequence (at 3), A085964 (at 4) to A085969 (at 9).
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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STATUS
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approved
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