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

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, 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, 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

Offset: 0

Author

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

Keywords

• cons
• nonn

Examples

			5.121028122527738383259898597063472005396598569391504803757141806973300...* 10^-7
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..1009
• X. Gourdon and P. Sebah, Some Constants from Number theory.
• R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=1, s=9), page 21.
• OEIS index to entries related to the (prime) zeta function.

Crossrefs

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).

Programs

Mathematica

digits = 1004;
nmax0 = 50; (* initial number of sum terms *)
dnmax = 10; (* nmax increment *)
dd = 10; (* precision excess *)
Clear[PrimeZeta41];
f[s_] := (1 + 2^-s)^-1 * DirichletBeta[s] Zeta[s]/Zeta[2 s];
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]&;
PrimeZeta41[9, nmax = nmax0];
PrimeZeta41[9, nmax += dnmax];
While[Abs[PrimeZeta41[9, nmax] - PrimeZeta41[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax];
PrimeZeta41[9] = PrimeZeta41[9, nmax];
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 *)

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

Formula

Zeta_Q(9) = Sum_{p in A002144} 1/p^9, where A002144 = {primes p == 1 mod 4};

= 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

Extensions

Edited by M. F. Hasler, Apr 26 2021