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%I #29 May 08 2021 08:37:31
%S 0,0,0,0,5,0,8,3,0,4,7,2,1,5,0,1,9,7,8,8,9,2,3,5,2,5,9,1,5,0,9,2,3,4,
%T 1,1,1,8,9,6,2,2,3,8,0,6,8,9,8,8,1,6,3,9,3,9,9,7,9,5,2,1,6,0,2,5,6,1,
%U 3,0,2,8,9,2,1,4,9,7,3,7,8,7,3,7,8,4,6,1,2,7,6,5,4,7,9,2,4,2,9,1,1,2,4,8,1
%N Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+3.
%H Jean-François Alcover, <a href="/A085998/b085998.txt">Table of n, a(n) for n = 0..1006</a>
%H P. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta Function Expansions of Classical Constants</a>, Unpublished manuscript. 1996.
%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/1008.2547">Table of ... Prime Zeta Modulo functions</a>, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=9), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_R(9) = Sum_{primes p == 3 (mod 4)} 1/p^9
%F = (1/2)*Sum_{n>=0} mobius(2*n+1) *log(b((2*n+1)*9))/(2*n+1),
%F where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.000050830472150197889235259150923411189622380689881639399795... ~ 5.08...*10^-5
%t digits = 1003;
%t nmax0 = 100;(* initial number of sum terms *)
%t dnmax = 10;(* nmax increment *)
%t dd = 10;(* precision excess *)
%t Clear[PrimeZeta43];
%t f[s_] := (1 - 2^(-s))*(Zeta[s]/DirichletBeta[s]);
%t PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd] &;
%t PrimeZeta43[9, nmax = nmax0];
%t PrimeZeta43[9, nmax += dnmax];
%t While[Abs[PrimeZeta43[9, nmax] - PrimeZeta43[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
%t PrimeZeta43[9] = PrimeZeta43[9, nmax];
%t Join[{0, 0, 0, 0}, RealDigits[PrimeZeta43[9], 10, digits][[1]]] (* _Jean-François Alcover_, Jun 22 2011, updated May 07 2021 *)
%o (PARI) A085998_upto(N=100)={localprec(N+3); digits((PrimeZeta43(9)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - _M. F. Hasler_, Apr 25 2021
%Y Cf. A085991 .. A085997 (Zeta_R(2..8)).
%Y Cf. A086039 (analog for primes 4k+1), A085969 (PrimeZeta(9)), A002145 (primes 4k+3).
%K cons,nonn
%O 0,5
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
%E Edited by _M. F. Hasler_, Apr 25 2021
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