|
%I #37 Jan 25 2024 07:52:35
%S 0,4,1,0,0,7,5,5,6,5,6,6,4,7,3,0,3,1,9,2,8,8,8,6,5,4,8,8,5,1,9,6,0,0,
%T 2,5,9,2,4,3,0,0,0,6,0,7,0,5,7,2,3,8,1,7,4,4,8,6,4,5,6,4,1,7,1,1,7,2,
%U 2,8,7,4,4,2,8,0,7,0,6,5,7,8,3,2,1,3,7,7,3,4,9,7,4,0,8,0,0,4,8,1,3,3,9,2,2
%N Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+3.
%H P. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta Function Expansions of Classical Constants</a>, Unpublished manuscript. 1996.
%H P. Fortuny Ayuso, J. M. Grau, and A. Oller-Marcen, <a href="http://arxiv.org/abs/1402.0333">A von Staudt-type formula for sum_{z in Z_n[i]} z^k</a>, arXiv:1402.0333 [math.NT], 2014.
%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...</a>, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, s=3, n=3), page 21.
%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F Zeta_R(3) = Sum_{primes p == 3 (mod 4)} 1/p^3
%F = (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*3))/(2*n+1),
%F where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
%e 0.04100755656647303192888654885196002592430006070572381744864564171...
%t b[x_] = (1 - 2^(-x))*(Zeta[x] / DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Prepend[ RealDigits[(1/2)* NSum[MoebiusMu[2n+1]* Log[b[(2n+1)*3]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* _Jean-François Alcover_, Jun 21 2011, updated Mar 14 2018 *)
%o (PARI) A085992_upto(N=100)={localprec(N+3); digits((PrimeZeta43(3)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - _M. F. Hasler_, Apr 25 2021
%Y Cf. A085991 .. A085998 (Zeta_R(2..9)).
%Y Cf. A086033 (analog for primes 4k+1), A085541 (PrimeZeta(3)), A002145 (primes 4k+3).
%K cons,nonn
%O 0,2
%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
%E Edited by _M. F. Hasler_, Apr 25 2021
|
