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A086033
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Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1.
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5
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0, 0, 8, 7, 5, 5, 0, 8, 2, 7, 3, 2, 9, 7, 0, 5, 0, 4, 4, 9, 4, 2, 2, 6, 7, 6, 5, 8, 1, 3, 7, 4, 6, 6, 7, 5, 0, 5, 1, 1, 1, 2, 0, 6, 1, 2, 2, 0, 4, 2, 5, 4, 7, 2, 4, 4, 0, 0, 2, 6, 3, 7, 4, 9, 8, 9, 9, 0, 8, 7, 1, 5, 1, 0, 0, 0, 5, 8, 9, 2, 9, 8, 0, 3, 4, 9, 6, 4, 6, 5, 5, 6, 2, 8, 9, 2, 5, 1, 2, 4, 1, 2, 8, 6, 8
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OFFSET
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0,3
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LINKS
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FORMULA
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Zeta_Q(3) = Sum_{p in A002144} 1/p^3 where A002144 = {primes p == 1 (mod 4)};
= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(3m)*zeta(3m)/zeta(6m)/(1+8^-m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021
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EXAMPLE
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0.008755082732970504494226765813746675051112061220425472440026374989908715100...
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MATHEMATICA
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a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 110; $MaxExtraPrecision = 470; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*3]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)
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PROG
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(PARI) A086033_upto(N=100)={localprec(N+3); digits((PrimeZeta41(3)+1)\.1^N)[^1]} \\ See A086032 for the function PrimeZeta41. - M. F. Hasler, Apr 24 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003
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STATUS
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approved
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