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

Author

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

Keywords

• cons
• nonn

Examples

			0.012843555610217553343622534619519018334553149771008458117126483020416...
		

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• P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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

Crossrefs

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

Cf. A086034 (analog for primes 4k+1), A085964 (PrimeZeta(4)), A002145 (primes 4k+3).

Programs

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 *)

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

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.

Extensions

Edited by M. F. Hasler, Apr 25 2021