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%I #23 May 08 2021 08:37:25
%S 0,0,0,0,6,4,2,5,0,9,6,3,6,6,4,7,7,3,7,9,1,1,0,1,8,1,9,1,3,8,0,4,3,5,
%T 7,6,5,9,8,9,8,4,5,4,5,5,4,6,9,7,8,8,1,5,0,5,2,8,9,8,5,6,6,2,5,8,4,3,
%U 8,9,8,4,5,2,0,0,9,7,7,4,5,3,2,3,9,4,4,7,4,5,8,2,6,4,7,0,4,5,7,0,1,1,9,4,4
%N Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1.
%H Jean-François Alcover, <a href="/A086036/b086036.txt">Table of n, a(n) for n = 0..1006</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=6), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_Q(6) = Sum_{p in A002144} 1/p^6 where A002144 = {primes p == 1 mod 4};
%F = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(6m)*zeta(6m)/zeta(12m)/(1+2^(-6m))) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021
%e 6.4250963664773791101819138043576598984545546978815052898566258438984520...*10^-5
%t digits = 1003; 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[(2n + 1)*6]]/(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}, RealDigits[g[m], 10, digits][[1]]] (* _Jean-François Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)
%o (PARI) A086036_upto(N=100)={localprec(N+3); digits((PrimeZeta41(6)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - _M. F. Hasler_, Apr 26 2021
%Y Cf. A085995 (same for primes 4k+3), A343626 (for primes 3k+1), A343616 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085966 (PrimeZeta(6)), A002144 (primes of the form 4k+1).
%K cons,nonn
%O 0,5
%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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