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A086033 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1. 5

%I

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

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

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

%N Decimal expansion of the prime zeta modulo function at 3 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, Section 3.2, constant P(m=4, n=1, s=3).

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

%F Zeta_Q(3) = Sum_{p in A002144} 1/p^3 where A002144 = {primes p == 1 (mod 4)};

%F = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(3m)*zeta(3m)/zeta(6m)/(1+8^-m))) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021

%F Equals A085541 - 1/2^3 - A085992. - _R. J. Mathar_, Apr 03 2011

%e 0.008755082732970504494226765813746675051112061220425472440026374989908715100...

%t a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 110; $MaxExtraPrecision = 470; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*3]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* _Jean-Fran├žois Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

%o (PARI) A086033_upto(N=100)={localprec(N+3);digits((PrimeZeta41(3)+1)\.1^N)[^1]} \\ See A086032 for the function PrimeZeta41. - _M. F. Hasler_, Apr 24 2021

%Y Cf. A085992 (same for primes 4k+3), A175645 (for primes 3k+1), A343613 (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 0,3

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

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Last modified August 5 14:57 EDT 2021. Contains 346469 sequences. (Running on oeis4.)