A085993 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+3.

0, 1, 2, 8, 4, 3, 5, 5, 5, 6, 1, 0, 2, 1, 7, 5, 5, 3, 3, 4, 3, 6, 2, 2, 5, 3, 4, 6, 1, 9, 5, 1, 9, 0, 1, 8, 3, 3, 4, 5, 5, 3, 1, 4, 9, 7, 7, 1, 0, 0, 8, 4, 5, 8, 1, 1, 7, 1, 2, 6, 4, 8, 3, 0, 2, 0, 4, 1, 6, 0, 7, 2, 9, 6, 9, 6, 8, 6, 4, 1, 7, 5, 7, 3, 5, 3, 1, 2, 7, 8, 6, 9, 8, 1, 7, 3, 2, 5, 3, 0, 7, 8, 0, 9, 9

Offset

0,3

Links

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

P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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=4), page 21.

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

Formula

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

= (1/2)*Sum_{n >= 0} mobius(2*n+1)*log(b((2*n+1)*4))/(2*n+1),

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

Example

0.012843555610217553343622534619519018334553149771008458117126483020416...

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)*4]]/(2n+1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]], 0][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

Prog

(PARI) A085993_upto(N=100)={localprec(N+3); digits((PrimeZeta43(4)+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. A086034 (analog for primes 4k+1), A085964 (PrimeZeta(4)), A002145 (primes 4k+3).

Sequence in context: A152626 A093823 A088154 * A010595 A109594 A329661

Adjacent sequences: A085990 A085991 A085992 * A085994 A085995 A085996

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