A085994 Decimal expansion of the prime zeta modulo function at 5 for primes of the form 4k+3.

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

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Keywords

• cons
• nonn

Examples

			0.004181543449702459614306334352814627154254543020852184353396741251345574...
		

Links

• P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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=5), page 21.
• OEIS index to entries related to the (prime) zeta function.

Crossrefs

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

Cf. A086035 (analog for primes 4k+1), A085965 (PrimeZeta(5)), A002145 (primes 4k+3).

Programs

Mathematica

b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 200; m = 40; Join[{0, 0}, RealDigits[(1/2)*NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*5]]/(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 *)

PARI

A085994_upto(N=100)={localprec(N+3); digits((PrimeZeta43(5)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021

Formula

Zeta_R(5) = Sum_{primes r == 3 mod 4} 1/p^5

= (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*5))/(2*n+1),

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