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%I #27 May 08 2021 08:37:19
%S 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,
%T 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,
%U 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
%N Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+1.
%H Jean-François Alcover, <a href="/A086038/b086038.txt">Table of n, a(n) for n = 0..1004</a>
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=1, s=8), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_Q(8) = Sum_{p in A002144} 1/p^8, where A002144 = {primes p == 1 (mod 4)};
%F = 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
%e 2.56137168039646980824843231247393644726060180729887066675459917474121... * 10^-6
%t digits = 1000; m0 = 50; dm = 10; dd = 10; Clear[f, g];
%t b[s_] := (1+2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;
%t f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[8(2n + 1)]]/(2n + 1);
%t g[m_] := g[m] = Sum[f[n], {n, 0, m}] ; g[m = m0]; g[m += dm];
%t While[Abs[g[m] - g[m - dm]] < 10^(-digits - dd), Print[m]; m += dm];
%t 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 *)
%o (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
%Y Cf. A085997 (same for primes 4k+3), A343628 (for primes 3k+1), A343618 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085968 (PrimeZeta(8)), A002144 (primes of the form 4k+1).
%K cons,nonn
%O 0,6
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003
%E Edited by _M. F. Hasler_, Apr 26 2021
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