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%I #35 Jan 25 2023 12:18:20
%S 1,4,8,4,3,3,6,5,6,4,6,7,0,0,7,8,2,8,2,2,5,8,6,5,0,7,7,4,9,0,7,1,1,3,
%T 7,1,8,8,7,5,5,5,8,4,1,7,4,4,8,0,6,8,8,9,4,4,2,5,0,7,5,0,8,0,5,5,2,9,
%U 8,2,0,0,3,1,9,7,6,8,2,2,9,3,0,6,4,3,0,9,8,6,8,5,0,6,7,2,4,6,9,0,9,3,5,0,7
%N Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+3.
%H Jean-François Alcover, <a href="/A085991/b085991.txt">Table of n, a(n) for n = 0..999</a>
%H P. Flajolet and I. Vardi, <a href="https://web.archive.org/web/20221130162351/http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta Function Expansions of Classical Constants</a>, Unpublished manuscript. 1996.
%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, section 3.2 constant P(m=4,n=3,s=2).
%F Zeta_R(2) = Sum_{primes p == 3 (mod 4)} 1/p^2
%F = (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*2))/(2*n+1),
%F where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.14843365646700782822586507749... = 1/3^2 + 1/7^2 + 1/11^2 + 1/19^2 + 1/23^2 + ...
%t digits = 1000; nmax0 = 500; dnmax = 10;
%t Clear[PrimeZeta43];
%t PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[(MoebiusMu[2n + 1] ((4n + 2) Log[2] + Log[((-1 + 2^(4n + 2)) Zeta[4n + 2])/(Zeta[4 n + 2, 1/4] - Zeta[4n + 2, 3/4])]))/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&;
%t PrimeZeta43[2, nmax = nmax0];
%t PrimeZeta43[2, nmax += dnmax];
%t While[Abs[PrimeZeta43[2, nmax] - PrimeZeta43[2, nmax - dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
%t PrimeZeta43[2] = PrimeZeta43[2, nmax];
%t RealDigits[PrimeZeta43[2], 10, digits][[1]] (* _Jean-François Alcover_, Jun 21 2011, updated May 06 2021 *)
%o (PARI) From _M. F. Hasler_, Apr 25 2021: (Start)
%o PrimeZeta43(s)={suminf(n=0, my(t=s+s*n*2); moebius(n*2+1)*log(zeta(t)/(zetahurwitz(t, 1/4)-zetahurwitz(t, 3/4))*(4^t-2^t))/(n*2+1))/2}
%o A085991_upto(N=100)={localprec(N+3); digits((PrimeZeta43(2)+1)\.1^N)[^1]} \\ (End)
%Y Cf. A086032 (analog for primes 4k+1), A085548 (PrimeZeta(2)), A002145.
%Y Cf. A085992 .. A085998 (Zeta_R(3..9)).
%K cons,nonn
%O 0,2
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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