A085994 - OEIS

%I #21 Aug 25 2021 15:28:24

%S 0,0,4,1,8,1,5,4,3,4,4,9,7,0,2,4,5,9,6,1,4,3,0,6,3,3,4,3,5,2,8,1,4,6,

%T 2,7,1,5,4,2,5,4,5,4,3,0,2,0,8,5,2,1,8,4,3,5,3,3,9,6,7,4,1,2,5,1,3,4,

%U 5,5,7,4,1,5,9,9,5,0,9,1,9,5,0,5,6,7,2,7,4,9,3,5,2,6,8,9,5,7,6,9,2,2,8,3,8

%N Decimal expansion of the prime zeta modulo function at 5 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, Unpublished manuscript. 1996.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 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=3, s=5), page 21.

%H <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.

%F Zeta_R(5) = Sum_{primes r == 3 mod 4} 1/p^5

%F   = (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*5))/(2*n+1),

%F   where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

%e 0.004181543449702459614306334352814627154254543020852184353396741251345574...

%t b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*5]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* _Jean-François Alcover_, Jun 22 2011, updated Mar 14 2018 *)

%o (PARI) A085994_upto(N=100)={localprec(N+3); digits((PrimeZeta43(5)+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. A086035 (analog for primes 4k+1), A085965 (PrimeZeta(5)), A002145 (primes 4k+3).

%K cons,nonn

%O 0,3

%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003