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A085991
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Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+3.
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14
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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, 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, 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
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OFFSET
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0,2
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LINKS
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FORMULA
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Zeta_R(2) = Sum_{primes p == 3 (mod 4)} 1/p^2
= (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*2))/(2*n+1),
where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
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EXAMPLE
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0.14843365646700782822586507749... = 1/3^2 + 1/7^2 + 1/11^2 + 1/19^2 + 1/23^2 + ...
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MATHEMATICA
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digits = 1000; nmax0 = 500; dnmax = 10;
Clear[PrimeZeta43];
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]&;
PrimeZeta43[2, nmax = nmax0];
PrimeZeta43[2, nmax += dnmax];
While[Abs[PrimeZeta43[2, nmax] - PrimeZeta43[2, nmax - dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta43[2] = PrimeZeta43[2, nmax];
RealDigits[PrimeZeta43[2], 10, digits][[1]] (* Jean-François Alcover, Jun 21 2011, updated May 06 2021 *)
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PROG
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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}
A085991_upto(N=100)={localprec(N+3); digits((PrimeZeta43(2)+1)\.1^N)[^1]} \\ (End)
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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 06 2003
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STATUS
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approved
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