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A085991 Decimal expansion of the Riemann zeta prime modulo function at 2 for primes of the form 4k+3. 9
1, 4, 8, 4, 3, 3, 6, 5, 6, 4, 6, 7, 0, 0, 7, 8, 2, 8, 2, 2, 5, 8, 6, 5, 0, 7, 7, 4, 9, 0, 7, 1, 1, 3, 7, 1, 8, 8, 7, 5, 5, 5, 8, 4, 1, 7, 4, 4, 8, 0, 6, 8, 8, 9, 4, 4, 2, 5, 0, 7, 5, 0, 8, 0, 5, 5, 2, 9, 8, 2, 0, 0, 3, 1, 9, 7, 6, 8, 2, 2, 9, 3, 0, 6, 4, 3, 0, 9, 8, 6, 8, 5, 0, 6, 7, 2, 4, 6, 9, 0, 9, 3, 5, 0, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547, section 3.2 constant P(m=4,n=3,s=2).

FORMULA

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

EXAMPLE

0.1484336564670078... = 1/3^2 + 1/7^2 + 1/11^2+ 1/19^2+ 1/23^2+..

MATHEMATICA

$MaxExtraPrecision = 120; m = 60; RealDigits[(1/2)* NSum[(MoebiusMu[2n + 1]*((4n + 2)*Log[2] + Log[((-1 + 2^(4n + 2))*Zeta[4n + 2])/(Zeta[4n + 2, 1/4] - Zeta[4n + 2, 3/4])]))/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120]][[1, 1 ;; 105]] (* Jean-Fran├žois Alcover, Jun 21 2011, after given formula *)

CROSSREFS

Cf. A086032, A085548, A002145.

Sequence in context: A014457 A092511 A045816 * A122110 A082632 A155874

Adjacent sequences:  A085988 A085989 A085990 * A085992 A085993 A085994

KEYWORD

cons,nonn

AUTHOR

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

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.