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A085992
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Decimal expansion of the Riemann zeta prime modulo function at 3 for primes of the form 4k+3.
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8
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0, 4, 1, 0, 0, 7, 5, 5, 6, 5, 6, 6, 4, 7, 3, 0, 3, 1, 9, 2, 8, 8, 8, 6, 5, 4, 8, 8, 5, 1, 9, 6, 0, 0, 2, 5, 9, 2, 4, 3, 0, 0, 0, 6, 0, 7, 0, 5, 7, 2, 3, 8, 1, 7, 4, 4, 8, 6, 4, 5, 6, 4, 1, 7, 1, 1, 7, 2, 2, 8, 7, 4, 4, 2, 8, 0, 7, 0, 6, 5, 7, 8, 3, 2, 1, 3, 7, 7, 3, 4, 9, 7, 4, 0, 8, 0, 0, 4, 8, 1, 3, 3, 9, 2, 2
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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LINKS
| P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
X. Gourdon and P. Sebah, Some Constants from Number theory.
R. J. Mathar, Table of Dirichlet L-series and.., arXiv:1008.2547, variable P(m=4,s=n=3), page 21.
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FORMULA
| Zeta_R(3) = Sum_{r prime=3 mod 4} 1/r^3 = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*3))/(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
| 0.0410075565664730...
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MATHEMATICA
| DirichletBeta[x_] = (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x;
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)*3]]/(2n+1), {n, 0, m},
AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]]
(* From Jean-François Alcover, Jun 21 2011 *)
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CROSSREFS
| Cf. A085991.
Sequence in context: A036877 A049763 A182878 * A117411 A161739 A094924
Adjacent sequences: A085989 A085990 A085991 * A085993 A085994 A085995
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KEYWORD
| cons,nonn
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AUTHOR
| Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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