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A086038
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Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+1.
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5
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0, 0, 0, 0, 0, 2, 5, 6, 1, 3, 7, 1, 6, 8, 0, 3, 9, 6, 4, 6, 9, 8, 0, 8, 2, 4, 8, 4, 3, 2, 3, 1, 2, 4, 7, 3, 9, 3, 6, 4, 4, 7, 2, 6, 0, 6, 0, 1, 8, 0, 7, 2, 9, 8, 8, 7, 0, 6, 6, 6, 7, 5, 4, 5, 9, 9, 1, 7, 4, 7, 4, 1, 2, 1, 1, 1, 8, 8, 8, 4, 8, 9, 3, 8, 8, 9, 7, 9, 8, 9, 1, 4, 8, 1, 7, 8, 0, 3, 0, 3, 0, 1, 3, 7, 6
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OFFSET
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0,6
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LINKS
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FORMULA
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Zeta_Q(8) = Sum_{p in A002144} 1/p^8, where A002144 = {primes p == 1 (mod 4)};
= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(8m)*zeta(8m)/zeta(16m)/(1+2^(-8m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021
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EXAMPLE
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2.56137168039646980824843231247393644726060180729887066675459917474121... * 10^-6
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MATHEMATICA
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digits = 1000; m0 = 50; dm = 10; dd = 10; Clear[f, g];
b[s_] := (1+2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;
f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[8(2n + 1)]]/(2n + 1);
g[m_] := g[m] = Sum[f[n], {n, 0, m}] ; g[m = m0]; g[m += dm];
While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
Join[{0, 0, 0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)
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PROG
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(PARI) A086038_upto(N=100)={localprec(N+3); digits((PrimeZeta41(8)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 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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EXTENSIONS
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STATUS
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approved
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