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%I #48 Feb 08 2024 07:13:13
%S 0,7,6,9,9,3,1,3,9,7,6,4,2,4,6,8,4,4,9,4,2,6,1,9,2,9,5,9,3,3,1,5,7,8,
%T 7,0,1,6,2,0,4,1,0,5,9,7,1,4,8,4,3,1,9,0,2,6,4,9,3,8,0,0,8,8,5,9,2,1,
%U 6,5,7,0,4,8,7,5,6,4,2,0,6,5,1,0,3,3,3,1,0,6,7,8,5,3,9,6,2,8,9,5,4,2,0,2,9
%N Decimal expansion of the prime zeta function at 4.
%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="/A085964/b085964.txt">Table of n, a(n) for n = 0..1603</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(4) = Sum_{p prime} 1/p^4 = Sum_{n>=1} mobius(n)*log(zeta(4*n))/n
%F Equals A086034 + A085993 + 1/16. - _R. J. Mathar_, Jul 22 2010
%F Equals Sum_{k>=1} 1/A030514(k). - _Amiram Eldar_, Jul 27 2020
%e 0.0769931397642468449426...
%t s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[4*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 *)
%t RealDigits[ PrimeZetaP[ 4], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 03 2014 *)
%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 [0]cat Reverse(IntegerToSequence(Floor(PrimeZeta(4,87)*10^105)));
%o // _Jason Kimberley_, Dec 30 2016
%o (PARI) sumeulerrat(1/p,4) \\ _Hugo Pfoertner_, Feb 03 2020
%Y Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), this sequence (at 4), A085965 (at 5) to A085969 (at 9).
%Y Cf. A013662, A030514, A242303.
%K cons,easy,nonn
%O 0,2
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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