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%I #45 Feb 13 2024 06:59:33
%S 0,3,5,7,5,5,0,1,7,4,8,3,9,2,4,2,5,7,1,3,2,8,1,8,2,4,2,5,3,8,8,5,5,7,
%T 1,1,1,3,1,6,9,7,2,7,6,7,2,6,6,5,1,3,3,1,6,9,0,0,9,2,6,7,4,8,2,3,9,7,
%U 5,8,3,4,2,7,4,7,2,7,9,3,1,3,6,6,0,7,2,8,0,6,4,7,0,3,7,6,7,9,5,0,8,9,6,3,9
%N Decimal expansion of the prime zeta function at 5.
%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="/A085965/b085965.txt">Table of n, a(n) for n = 0..1702</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%F P(5) = Sum_{p prime} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.
%F Equals 1/2^5 + A085994 + A086035. - _R. J. Mathar_, Jul 14 2012
%F Equals Sum_{k>=1} 1/A050997(k). - _Amiram Eldar_, Jul 27 2020
%e 0.0357550174839242571328...
%t s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n=200]; While[s[n] != s[n-100], n = n+100]; s[n] (* _Jean-François Alcover_, Feb 14 2013, from 1st formula *)
%t RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 03 2014 *)
%o (Magma) R := RealField(106);
%o PrimeZeta := func<k,N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R,k*n)): n in[1..N]]>;
%o [0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5,69)*10^105)));
%o // _Jason Kimberley_, Dec 30 2016
%o (PARI) sumeulerrat(1/p,5) \\ _Hugo Pfoertner_, Feb 03 2020
%Y Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), this sequence (at 5), A085966 (at 6) to A085969 (at 9).
%Y Cf. A013663, A050997, A242304.
%K cons,easy,nonn
%O 0,2
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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