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A085993 Decimal expansion of the Riemann zeta prime modulo function at 4 for primes of the form 4k+3. 6

%I

%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 Riemann zeta prime 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, Unpublished manuscript. 1996.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>.

%F Zeta_R(4) = Sum_{r prime=3 mod 4} 1/r^4 = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*4))/(2*n+1), where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

%e 0.0128435556102175...

%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-Fran├žois Alcover_, Jun 22 2011, updated Mar 14 2018 *)

%Y Cf. A085991, A085992.

%K cons,nonn

%O 0,3

%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

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