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A085993
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Decimal expansion of the Riemann zeta prime modulo function at 4 for primes of the form 4k+3.
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6
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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, 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, 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
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..104.
P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
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FORMULA
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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.
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EXAMPLE
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0.0128435556102175...
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A085991, A085992.
Sequence in context: A152626 A093823 A088154 * A010595 A109594 A329661
Adjacent sequences: A085990 A085991 A085992 * A085994 A085995 A085996
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KEYWORD
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cons,nonn
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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STATUS
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approved
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