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A085969
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Decimal expansion of the prime zeta function at 9.
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25
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2, 0, 0, 4, 4, 6, 7, 5, 7, 4, 9, 6, 2, 4, 5, 0, 6, 6, 3, 0, 7, 3, 5, 8, 5, 1, 4, 0, 7, 8, 3, 1, 1, 7, 5, 3, 6, 8, 2, 2, 9, 2, 0, 3, 4, 9, 7, 3, 8, 5, 7, 8, 5, 5, 0, 2, 7, 5, 5, 9, 1, 7, 3, 9, 7, 9, 3, 7, 2, 6, 2, 5, 1, 5, 1, 9, 9, 7, 7, 1, 9, 0, 8, 1, 3, 5, 5, 3, 4, 3, 4, 0, 1, 2, 4, 4, 1, 7, 7, 2, 4, 3, 0
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OFFSET
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-2,1
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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 07 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(9) = Sum_{p prime} 1/p^9 = Sum_{n=1..inf} mobius(n)*log(zeta(9*n))/n.
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EXAMPLE
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0.0020044675749624506630...
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MATHEMATICA
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pz9[n_] := pz9[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[9*k]]/k, {k, 1, n}] // RealDigits[#, 10, 103]& // First]; pz9[100]; pz9[n = 200]; While[pz9[n] != pz9[n - 100], n = n + 100]; pz9[n] (* Jean-François Alcover, Feb 14 2013, from formula *)
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PROG
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(Magma) R := RealField(106);
PrimeZeta := func<k, N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)): n in[1..N]]>;
[0, 0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(9, 38)*10^105)));
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CROSSREFS
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KEYWORD
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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EXTENSIONS
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STATUS
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approved
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