A086035 - OEIS

%I #31 May 08 2021 08:36:59

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

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

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

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

%H Jean-François Alcover, <a href="/A086035/b086035.txt">Table of n, a(n) for n = 0..1006</a>

%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=5), page 21.

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

%F Zeta_Q(5) = Sum_{p in A002144} 1/p^5  where  A002144 = {primes p == 1 (mod 4)};

%F   = Sum_{odd m > 0} mu(m)/2m *log(DirichletBeta(5m)*zeta(5m)/zeta(10m)/(1+2^(-5m))) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021

%e 0.0003234740342217975185119081860410839774427337057991473366959335726302601...

%t digits = 1004; m0 = 50; dm = 10; dd = 10; Clear[f, g];

%t b[s_] := (1+2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2s] // N[#, digits+dd]&;

%t f[n_] := f[n] = (1/2) MoebiusMu[2n + 1] Log[b[5(2n + 1)]]/(2n + 1);

%t g[m_] := g[m] = Sum[f[n], {n, 0, m}]; g[m = m0]; g[m += dm];

%t While[Abs[g[m] - g[m-dm]] < 10^(-digits-dd), Print[m]; m += dm];

%t Join[{0, 0, 0}, RealDigits[g[m], 10, digits][[1]]] (* _Jean-François Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 08 2021 *)

%o (PARI) A086035_upto(N=100)={localprec(N+3); digits((PrimeZeta41(5)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - _M. F. Hasler_, Apr 26 2021

%Y Cf. A085994 (same for primes 4k+3), A343625 (for primes 3k+1), A343615 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085965 (PrimeZeta(5)), 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