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A085997 Decimal expansion of the Riemann zeta prime modulo function at 8 for primes of the form 4k+3. 1
0, 0, 0, 1, 5, 2, 5, 9, 3, 9, 9, 4, 8, 3, 7, 4, 3, 4, 0, 9, 0, 7, 1, 5, 1, 9, 0, 7, 1, 0, 3, 7, 0, 6, 0, 6, 5, 8, 6, 5, 2, 9, 8, 8, 3, 9, 1, 0, 2, 6, 4, 4, 4, 2, 1, 3, 0, 3, 6, 5, 9, 3, 4, 0, 8, 2, 5, 5, 3, 8, 8, 9, 1, 9, 5, 8, 8, 9, 9, 5, 5, 4, 6, 7, 1, 9, 4, 2, 9, 3, 6, 5, 7, 1, 2, 6, 2, 8, 3, 1, 4, 1, 2, 7, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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.

FORMULA

Zeta_R(8) = Sum_{r prime=3 mod 4} 1/r^8 = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*8))/(2*n+1), where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

EXAMPLE

0.0001525939948374...

MATHEMATICA

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 320; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*8]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-Fran├žois Alcover, Jun 22 2011, updated Mar 14 2018 *)

CROSSREFS

Cf. A085991, A085992, A085993, A085994, A085995, A085996.

Sequence in context: A142702 A236184 A201530 * A071546 A154649 A100040

Adjacent sequences:  A085994 A085995 A085996 * A085998 A085999 A086000

KEYWORD

cons,nonn

AUTHOR

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

STATUS

approved

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)