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A085992 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+3. 5
0, 4, 1, 0, 0, 7, 5, 5, 6, 5, 6, 6, 4, 7, 3, 0, 3, 1, 9, 2, 8, 8, 8, 6, 5, 4, 8, 8, 5, 1, 9, 6, 0, 0, 2, 5, 9, 2, 4, 3, 0, 0, 0, 6, 0, 7, 0, 5, 7, 2, 3, 8, 1, 7, 4, 4, 8, 6, 4, 5, 6, 4, 1, 7, 1, 1, 7, 2, 2, 8, 7, 4, 4, 2, 8, 0, 7, 0, 6, 5, 7, 8, 3, 2, 1, 3, 7, 7, 3, 4, 9, 7, 4, 0, 8, 0, 0, 4, 8, 1, 3, 3, 9, 2, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..104.

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.

P. Fortuny Ayuso, J. M. Grau, and A. Oller-Marcen, A von Staudt-type formula for sum_{z in Z_n[i]} z^k, arXiv:1402.0333 [math.NT], 2014.

X. Gourdon and P. Sebah, Some Constants from Number theory.

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions..., arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, s=3, n=3), page 21.

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

FORMULA

Zeta_R(3) = Sum_{primes p == 3 (mod 4)} 1/p^3

  = (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*3))/(2*n+1),

  where b(x) = (1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.

EXAMPLE

0.04100755656647303192888654885196002592430006070572381744864564171...

MATHEMATICA

b[x_] = (1 - 2^(-x))*(Zeta[x] / DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Prepend[ RealDigits[(1/2)* NSum[MoebiusMu[2n+1]* Log[b[(2n+1)*3]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* Jean-Fran├žois Alcover, Jun 21 2011, updated Mar 14 2018 *)

PROG

A085992_upto(N=100)={localprec(N+3); digits((PrimeZeta43(3)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

CROSSREFS

Cf. A085991 .. A085998 (Zeta_R(2..9)).

Cf. A086033 (analog for primes 4k+1), A085541 (PrimeZeta(3)), A002145 (primes 4k+3).

Sequence in context: A297785 A334702 A345300 * A117411 A161739 A291574

Adjacent sequences:  A085989 A085990 A085991 * A085993 A085994 A085995

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

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Last modified July 29 08:28 EDT 2021. Contains 346340 sequences. (Running on oeis4.)