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A086036 Decimal expansion of the prime zeta modulo function at 6 for primes of the form 4k+1. 5
0, 0, 0, 0, 6, 4, 2, 5, 0, 9, 6, 3, 6, 6, 4, 7, 7, 3, 7, 9, 1, 1, 0, 1, 8, 1, 9, 1, 3, 8, 0, 4, 3, 5, 7, 6, 5, 9, 8, 9, 8, 4, 5, 4, 5, 5, 4, 6, 9, 7, 8, 8, 1, 5, 0, 5, 2, 8, 9, 8, 5, 6, 6, 2, 5, 8, 4, 3, 8, 9, 8, 4, 5, 2, 0, 0, 9, 7, 7, 4, 5, 3, 2, 3, 9, 4, 4, 7, 4, 5, 8, 2, 6, 4, 7, 0, 4, 5, 7, 0, 1, 1, 9, 4, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..1006

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=6), page 21.

OEIS index to entries related to the (prime) zeta function.

FORMULA

Zeta_Q(6) = Sum_{p in A002144} 1/p^6  where  A002144 = {primes p == 1 mod 4};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(6m)*zeta(6m)/zeta(12m)/(1+2^(-6m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

EXAMPLE

6.4250963664773791101819138043576598984545546978815052898566258438984520...*10^-5

MATHEMATICA

digits = 1003; m0 = 50; dm = 10; dd = 10; Clear[f, g];

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

f[n_] := f[n] = (1/2) MoebiusMu[2n + 1]*Log[b[(2n + 1)*6]]/(2n + 1);

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

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

Join[{0, 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 *)

PROG

(PARI) A086036_upto(N=100)={localprec(N+3); digits((PrimeZeta41(6)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

CROSSREFS

Cf. A085995 (same for primes 4k+3), A343626 (for primes 3k+1), A343616 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085966 (PrimeZeta(6)), A002144 (primes of the form 4k+1).

Sequence in context: A028975 A145011 A173625 * A345644 A019849 A244685

Adjacent sequences:  A086033 A086034 A086035 * A086037 A086038 A086039

KEYWORD

cons,nonn

AUTHOR

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

EXTENSIONS

Edited by M. F. Hasler, Apr 26 2021

STATUS

approved

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)