cp's OEIS Frontend

A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.

%I A086039 #28 Jun 04 2025 10:24:18
%S A086039 0,0,0,0,0,0,5,1,2,1,0,2,8,1,2,2,5,2,7,7,3,8,3,8,3,2,5,9,8,9,8,5,9,7,
%T A086039 0,6,3,4,7,2,0,0,5,3,9,6,5,9,8,5,6,9,3,9,1,5,0,4,8,0,3,7,5,7,1,4,1,8,
%U A086039 0,6,9,7,3,3,0,0,2,2,5,9,8,4,5,2,9,6,7,4,2,5,7,7,9,2,2,0,0,1,2,6,5,9,9,4,8
%N A086039 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+1.
%H A086039 Jean-François Alcover, <a href="/A086039/b086039.txt">Table of n, a(n) for n = 0..1009</a>
%H A086039 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.
%H A086039 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=9), page 21.
%H A086039 <a href="/index/Z#zeta_function">OEIS index to entries related to the (prime) zeta function</a>.
%F A086039 Zeta_Q(9) = Sum_{p in A002144} 1/p^9, where A002144 = {primes p == 1 mod 4};
%F A086039   = Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(9m)*zeta(9m)/zeta(18m)/(1 + 2^(-9m))) [using Gourdon & Sebah, Theorem 11]. - _M. F. Hasler_, Apr 26 2021
%e A086039 5.121028122527738383259898597063472005396598569391504803757141806973300...* 10^-7
%t A086039 digits = 1004;
%t A086039 nmax0 = 50; (* initial number of sum terms *)
%t A086039 dnmax = 10; (* nmax increment *)
%t A086039 dd = 10; (* precision excess *)
%t A086039 Clear[PrimeZeta41];
%t A086039 f[s_] := (1 + 2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2 s];
%t A086039 PrimeZeta41[s_, nmax_] := PrimeZeta41[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd]&;
%t A086039 PrimeZeta41[9, nmax = nmax0];
%t A086039 PrimeZeta41[9, nmax += dnmax];
%t A086039 While[Abs[PrimeZeta41[9, nmax] - PrimeZeta41[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
%t A086039 PrimeZeta41[9] = PrimeZeta41[9, nmax];
%t A086039 Join[{0, 0, 0, 0, 0, 0}, RealDigits[PrimeZeta41[9], 10, digits][[1]]] (* _Jean-François Alcover_, Jun 24 2011, after X. Gourdon and P. Sebah, updated May 07 2021 *)
%o A086039 (PARI) A086039_upto(N=100)={localprec(N+3); digits((PrimeZeta41(9)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - _M. F. Hasler_, Apr 26 2021
%Y A086039 Cf. A085998 (same for primes 4k+3), A343629 (for primes 3k+1), A343619 (for primes 3k+2), A086032 - A086038 (for 1/p^2, ..., 1/p^8), A085969 (PrimeZeta(9)), A002144 (primes of the form 4k+1).
%K A086039 cons,nonn
%O A086039 0,7
%A A086039 Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003
%E A086039 Edited by _M. F. Hasler_, Apr 26 2021