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%I #29 Jan 25 2024 07:52:47
%S 0,1,2,8,4,3,5,5,5,6,1,0,2,1,7,5,5,3,3,4,3,6,2,2,5,3,4,6,1,9,5,1,9,0,
%T 1,8,3,3,4,5,5,3,1,4,9,7,7,1,0,0,8,4,5,8,1,1,7,1,2,6,4,8,3,0,2,0,4,1,
%U 6,0,7,2,9,6,9,6,8,6,4,1,7,5,7,3,5,3,1,2,7,8,6,9,8,1,7,3,2,5,3,0,7,8,0,9,9
%N Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+3.
%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, Unpublished manuscript. 1996.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=4), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_R(4) = Sum_{primes p == 3 mod 4} 1/p^4
%F = (1/2)*Sum_{n >= 0} mobius(2*n+1)*log(b((2*n+1)*4))/(2*n+1),
%F where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.012843555610217553343622534619519018334553149771008458117126483020416...
%t b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Prepend[ RealDigits[ (1/2)*NSum[ MoebiusMu[2n+1]* Log[b[(2n+1)*4]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* _Jean-François Alcover_, Jun 22 2011, updated Mar 14 2018 *)
%o (PARI) A085993_upto(N=100)={localprec(N+3); digits((PrimeZeta43(4)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - _M. F. Hasler_, Apr 25 2021
%Y Cf. A085991 .. A085998 (Zeta_R(2..9)).
%Y Cf. A086034 (analog for primes 4k+1), A085964 (PrimeZeta(4)), A002145 (primes 4k+3).
%K cons,nonn
%O 0,3
%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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