A085998 Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+3.
0, 0, 0, 0, 5, 0, 8, 3, 0, 4, 7, 2, 1, 5, 0, 1, 9, 7, 8, 8, 9, 2, 3, 5, 2, 5, 9, 1, 5, 0, 9, 2, 3, 4, 1, 1, 1, 8, 9, 6, 2, 2, 3, 8, 0, 6, 8, 9, 8, 8, 1, 6, 3, 9, 3, 9, 9, 7, 9, 5, 2, 1, 6, 0, 2, 5, 6, 1, 3, 0, 2, 8, 9, 2, 1, 4, 9, 7, 3, 7, 8, 7, 3, 7, 8, 4, 6, 1, 2, 7, 6, 5, 4, 7, 9, 2, 4, 2, 9, 1, 1, 2, 4, 8, 1
Offset: 0
Examples
0.000050830472150197889235259150923411189622380689881639399795... ~ 5.08...*10^-5
Links
- Jean-François Alcover, Table of n, a(n) for n = 0..1006
- P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
- X. Gourdon and P. Sebah, Some Constants from Number theory.
- R. J. Mathar, Table of ... Prime Zeta Modulo functions, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=9), page 21.
- OEIS index to entries related to the (prime) zeta function.
Crossrefs
Programs
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Mathematica
digits = 1003; nmax0 = 100;(* initial number of sum terms *) dnmax = 10;(* nmax increment *) dd = 10;(* precision excess *) Clear[PrimeZeta43]; f[s_] := (1 - 2^(-s))*(Zeta[s]/DirichletBeta[s]); PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[MoebiusMu[2 n + 1]*Log[f[(2 n + 1)*9]]/(2 n + 1), {n, 0, nmax}] // N[#, digits + dd] &; PrimeZeta43[9, nmax = nmax0]; PrimeZeta43[9, nmax += dnmax]; While[Abs[PrimeZeta43[9, nmax] - PrimeZeta43[9, nmax - dnmax]] > 10^-(digits + dd), Print["nmax = ", nmax]; nmax += dnmax]; PrimeZeta43[9] = PrimeZeta43[9, nmax]; Join[{0, 0, 0, 0}, RealDigits[PrimeZeta43[9], 10, digits][[1]]] (* Jean-François Alcover, Jun 22 2011, updated May 07 2021 *) -
PARI
A085998_upto(N=100)={localprec(N+3); digits((PrimeZeta43(9)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021
Formula
Zeta_R(9) = Sum_{primes p == 3 (mod 4)} 1/p^9
= (1/2)*Sum_{n>=0} mobius(2*n+1) *log(b((2*n+1)*9))/(2*n+1),
where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.
Extensions
Edited by M. F. Hasler, Apr 25 2021