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%I #20 Apr 26 2021 01:53:22
%S 0,0,1,3,8,0,8,3,5,8,8,6,9,7,1,7,3,9,1,6,3,0,3,1,8,5,4,1,2,8,0,1,5,8,
%T 2,2,6,1,0,6,0,1,3,9,6,3,2,7,5,6,5,4,2,9,6,8,0,2,6,4,8,0,2,5,7,8,5,3,
%U 0,7,5,2,2,2,7,0,7,4,6,9,1,3,4,7,9,1,5,6,0,4,2,5,1,7,1,0,1,6,6,0,1,6,8,7,8
%N Decimal expansion of the prime zeta modulo function at 6 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, Unpublished manuscript. 1996.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 Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=6), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_R(6) = Sum_{p in A002145} 1/p^6 where A002145 = {primes p == 3 (mod 4)},
%F = (1/2)*Sum_{n >= 0} möbius(2*n+1)*log(b((2*n+1)*6))/(2*n+1),
%F where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.0013808358869717391630318541280158226106013963275654296802648025785307522...
%t b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 250; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*6]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* _Jean-François Alcover_, Jun 22 2011, updated Mar 14 2018 *)
%o (PARI) A085995_upto(N=100)={localprec(N+3); digits((PrimeZeta43(6)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - _M. F. Hasler_, Apr 25 2021
%Y Cf. A002145 (primes 4k+3), A001014 (n^6), A085966 (PrimeZeta(6)).
%Y Cf. A085991 - A085998 (Zeta_R(2..9): same for 1/p^2, ..., 1/p^9), A086036 (same for primes 4k+1), A343626 (for primes 3k+1), A343616 (for primes 3k+2).
%K cons,nonn
%O 0,4
%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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