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%I #15 Apr 28 2021 23:02:55
%S 0,0,0,4,5,8,5,1,4,4,0,7,5,3,3,7,9,7,2,6,6,8,7,3,1,1,2,1,4,7,2,8,2,2,
%T 1,5,1,5,3,3,6,2,7,2,2,1,3,5,7,4,4,4,6,1,4,5,0,2,7,9,2,6,4,7,2,3,9,7,
%U 3,2,9,5,0,1,1,5,1,2,7,7,2,8,9,8,9,9,2,7,1,8,0,7,7,6,4,5,3,9,2,5,8,9,3,5,3
%N Decimal expansion of the prime zeta modulo function at 7 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, Unpublished manuscript. 1996.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, value P(m=4, n=3, s=7), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_R(7) = Sum_{primes p == 3 mod 4} 1/p^7
%F = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*7))/(2*n+1),
%F where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.0004585144075337972668731121472822151533627221357444614502792647239732950115...
%t b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 275; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*7]]/(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) A085996_upto(N=100)={localprec(N+3); digits((PrimeZeta43(7)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - _M. F. Hasler_, Apr 25 2021
%Y Cf. A086037 (analog for primes 4k+1), A085967 (PrimeZeta(7)), A002145 (primes 4k+3).
%Y Cf. A085991 .. A085998 (Zeta_R(2..9)).
%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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