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A085998
Decimal expansion of the prime zeta modulo function at 9 for primes of the form 4k+3.
10
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
0,5
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P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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.
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.
EXAMPLE
0.000050830472150197889235259150923411189622380689881639399795... ~ 5.08...*10^-5
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 *)
PROG
(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
CROSSREFS
Cf. A085991 .. A085997 (Zeta_R(2..8)).
Cf. A086039 (analog for primes 4k+1), A085969 (PrimeZeta(9)), A002145 (primes 4k+3).
Sequence in context: A011510 A021667 A078119 * A094886 A371506 A371507
KEYWORD
cons,nonn
AUTHOR
Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
EXTENSIONS
Edited by M. F. Hasler, Apr 25 2021
STATUS
approved