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A085994
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Decimal expansion of the Riemann zeta prime modulo function at 5 for primes of the form 4k+3.
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4
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0, 0, 4, 1, 8, 1, 5, 4, 3, 4, 4, 9, 7, 0, 2, 4, 5, 9, 6, 1, 4, 3, 0, 6, 3, 3, 4, 3, 5, 2, 8, 1, 4, 6, 2, 7, 1, 5, 4, 2, 5, 4, 5, 4, 3, 0, 2, 0, 8, 5, 2, 1, 8, 4, 3, 5, 3, 3, 9, 6, 7, 4, 1, 2, 5, 1, 3, 4, 5, 5, 7, 4, 1, 5, 9, 9, 5, 0, 9, 1, 9, 5, 0, 5, 6, 7, 2, 7, 4, 9, 3, 5, 2, 6, 8, 9, 5, 7, 6, 9, 2, 2, 8, 3, 8
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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| P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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.
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FORMULA
| Zeta_R(5) = Sum_{r prime=3 mod 4} 1/r^5 = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*5))/(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
| 0.0041815434497024...
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MATHEMATICA
| DirichletBeta[s_] = (Zeta[s, 1/4] - Zeta[s, 3/4])/4^s; b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]);
$MaxExtraPrecision = 200; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*5]]/(2n + 1), {n, 0, m},
AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]]
(* From Jean-François Alcover, Jun 22 2011 *)
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CROSSREFS
| Cf. A085991, A085992, A085993.
Sequence in context: A143320 A089146 A101512 * A179836 A040019 A019768
Adjacent sequences: A085991 A085992 A085993 * A085995 A085996 A085997
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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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