%I #38 May 06 2021 22:13:49
%S 0,5,3,8,1,3,7,6,3,5,7,4,0,5,7,6,7,0,2,8,0,6,7,8,2,8,7,3,4,1,5,3,6,5,
%T 6,2,2,8,5,6,7,5,5,0,1,4,9,5,0,8,5,5,3,2,2,9,3,9,1,1,4,2,2,2,9,5,8,6,
%U 6,8,2,7,0,4,4,1,4,2,6,4,5,1,4,2,5,2,6,5,5,7,5,0,4,2,3,4,3,8,9,1,2,9,2,9,8
%N Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+1.
%H Jean-François Alcover, <a href="/A086032/b086032.txt">Table of n, a(n) for n = 0..1004</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, section 3.2, constant P(m=4,n=1,s=2).
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_Q(2) = Sum_{p in A002144} 1/p^2, where A002144 = {primes p == 1 (mod 4)}.
%F Equals A085548 - 1/4 - A085991. - R. J. Mathar, Apr 03 2011
%F Zeta_Q(2) = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(2m)*zeta(2m)/zeta(4m)/(1 + 4^-m)) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021
%e 0.053813763574057670280678287341536562285675501495085532293911422295866827...
%t digits = 1004; nmax0 = 100; dnmax = 10;
%t Clear[PrimeZeta41];
%t f[s_] := (1 + 2^-s)^-1*DirichletBeta[s] Zeta[s]/Zeta[2s];
%t PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2n + 1]* Log[f[(2n + 1)*2]]/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&;
%t PrimeZeta41[2, nmax = nmax0];
%t PrimeZeta41[2, nmax += dnmax];
%t While[Abs[PrimeZeta41[2, nmax] - PrimeZeta41[2, nmax - dnmax]] > 10^-(digits + 5), Print["nmax = ", nmax]; nmax += dnmax];
%t PrimeZeta41[2] = PrimeZeta41[2, nmax];
%t Join[{0}, RealDigits[PrimeZeta41[2], 10, digits][[1]]] (* _Jean-François Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 06 2021 *)
%o (PARI) From _M. F. Hasler_, Apr 24 2021: (Start)
%o PrimeZeta41(s)={suminf(n=0, my(t=s+s*n*2); moebius(2*n+1)*log(zeta(t)/zeta(2*t)*(zetahurwitz(t,1/4)-zetahurwitz(t,3/4))/(4^t+2^t))/(4*n+2))}
%o A086032_upto(N=100)={localprec(N+3);digits((PrimeZeta41(2)+1)\.1^N)[^1]} \\ (End)
%Y Cf. A085991 (same for primes 4k+3), A175644 (for primes 3k+1), A343612 (for primes 3k+2), A086033 - A086039 (for 1/p^3, ..., 1/p^9), A085548 (PrimeZeta(2)), A002144 (primes 4k+1).
%K cons,nonn
%O 0,2
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003