A086034 - OEIS

%I #18 Apr 26 2021 05:53:05

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

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

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

%N Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+1.

%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=1, s=4), page 21.

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

%F Zeta_Q(4) = Sum_{p in A002144} 1/p^4  where  A002144 = {primes p == 1 mod 4};

%F   = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(4m)*zeta(4m)/zeta(8m)/(1+2^(-4m)))[using Gourdon & Sebah, Theorem 11] - _M. F. Hasler_, Apr 26 2021.

%e 0.0016495841540292915989967613136388518274879099438347321478115258388...

%t a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 680; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*4]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* _Jean-François Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

%o (PARI) A086034_upto(N=100)={localprec(N+3); digits((PrimeZeta41(4)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - _M. F. Hasler_, Apr 26 2021

%Y Cf. A085993 (same for primes 4k+3), A343624 (for primes 3k+1), A343614 (for primes 3k+2), A086032 - A086039 (for 1/p^2, ..., 1/p^9), A085541 (PrimeZeta(3)), A002144 (primes of the form 4k+1).

%K cons,nonn

%O 0,4

%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

%E Edited by _M. F. Hasler_, Apr 26 2021