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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 OFFSET 0,2 LINKS P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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-François Alcover, Jun 21 2011, after given formula *) CROSSREFS Cf. A086032, A085548, A002145. Sequence in context: A014457 A092511 A045816 * A122110 A082632 A296481 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 April 5 16:24 EDT 2020. Contains 333245 sequences. (Running on oeis4.)