A085997 Decimal expansion of the prime zeta modulo function at 8 for primes of the form 4k+3.

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

Offset

0,5

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P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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 Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=4, n=3, s=8), page 21.

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

Formula

Zeta_R(8) = Sum_{primes p == 3 mod 4} 1/p^8

= (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*8))/(2*n+1),

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

Example

0.000152593994837434090715190710370606586529883910264442130365934082553889...

Mathematica

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 320; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*8]]/(2n + 1), {n, 0, m}, AccuracyGoal -> 120, NSumTerms -> m, PrecisionGoal -> 120, WorkingPrecision -> 120] ][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 22 2011, updated Mar 14 2018 *)

Prog

(PARI) A085997_upto(N=100)={localprec(N+3); digits((PrimeZeta43(8)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Crossrefs

Cf. A086038 (analog for primes 4k+1), A085968 (PrimeZeta(8)), A002145 (primes 4k+3).

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

Sequence in context: A142702 A236184 A201530 * A071546 A154649 A100040

Adjacent sequences: A085994 A085995 A085996 * A085998 A085999 A086000

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