A086037 - OEIS

%I #23 Apr 26 2021 05:52:37

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

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

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

%N Decimal expansion of the prime zeta modulo function at 7 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=7), page 21.

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

%F Zeta_Q(7) = Sum_{p in A002144} 1/p^7  where  A002144 = {primes p == 1 mod 4};

%F   = Sum_{odd m > 0} mu(m)/2m*log(DirichletBeta(7m)*zeta(7m)/zeta(14m)/(1+2^(-7m))) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021

%e 1.2818448599795268251026582166507935820606749563344794362656914682... * 10^-5

%t a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 120; $MaxExtraPrecision = 1200; Join[{0, 0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*7]]/(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) A086037_upto(N=100)={localprec(N+3); digits((PrimeZeta41(7)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - _M. F. Hasler_, Apr 26 2021

%Y Cf. A085996 (same for primes 4k+3), A343627 (for primes 3k+1), A343617 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085967 (PrimeZeta(7)), A002144 (primes of the form 4k+1).

%K cons,nonn

%O 0,6

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

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