cp's OEIS Frontend

A085967 Decimal expansion of the prime zeta function at 7.

%I A085967 #42 Feb 16 2025 08:32:50
%S A085967 0,0,8,2,8,3,8,3,2,8,5,6,1,3,3,5,9,2,5,3,5,1,2,4,1,3,8,7,2,9,4,4,8,7,
%T A085967 2,3,0,8,9,1,8,3,3,2,8,8,8,5,3,0,7,8,0,6,2,4,4,6,4,1,9,2,1,6,3,8,6,5,
%U A085967 5,4,8,9,4,1,1,0,0,7,8,5,8,1,8,4,3,1,6,6,1,3,4,1,8,1,9,1,8,2,0,0,4,3,2,8,1
%N A085967 Decimal expansion of the prime zeta function at 7.
%C A085967 Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - _Jason Kimberley_, Jan 07 2017
%D A085967 Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D A085967 J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
%H A085967 Jason Kimberley, <a href="/A085967/b085967.txt">Table of n, a(n) for n = 0..1902</a>
%H A085967 Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, Preprint, 1998.
%H A085967 Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H A085967 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>
%H A085967 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 A085967 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%F A085967 P(7) = Sum_{p prime} 1/p^7 = Sum_{n>=1} mobius(n)*log(zeta(7*n))/n.
%F A085967 Equals Sum_{k>=1} 1/A092759(k). - _Amiram Eldar_, Jul 27 2020
%e A085967 0.0082838328561335925351...
%t A085967 s[n_] := s[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[7*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104] & // First]; s[100]; s[n = 200]; While[ s[n] != s[n - 100], n = n + 100]; s[n] (* _Jean-François Alcover_, Feb 14 2013 *)
%t A085967 RealDigits[ PrimeZetaP[ 7], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 03 2014 *)
%o A085967 (Magma) R := RealField(106);
%o A085967 PrimeZeta := func<k,N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R,k*n)): n in[1..N]]>;
%o A085967 [0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(7,47)*10^105)));
%o A085967 // _Jason Kimberley_, Dec 30 2016
%o A085967 (PARI) sumeulerrat(1/p,7) \\ _Hugo Pfoertner_, Feb 03 2020
%Y A085967 Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085966 (at 6), this sequence (at 7), A085968 (at 8), A085969 (at 9).
%Y A085967 Cf. A013665, A092759.
%K A085967 cons,easy,nonn
%O A085967 0,3
%A A085967 Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003