A086033 Decimal expansion of the prime zeta modulo function at 3 for primes of the form 4k+1.

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

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

Keywords

• cons
• nonn

Examples

			0.008755082732970504494226765813746675051112061220425472440026374989908715100...
		

Links

• 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, Section 3.2, constant P(m=4, n=1, s=3).
• OEIS index to entries related to the (prime) zeta function.

Crossrefs

Cf. A085992 (same for primes 4k+3), A175645 (for primes 3k+1), A343613 (for primes 3k+2), A086032, ..., A086039 (for 1/p^2, ..., 1/p^9), A085541 (PrimeZeta(3)), A002144 (primes of the form 4k+1).

Programs

Mathematica

a[s_] = (1 + 2^-s)^-1* DirichletBeta[s] Zeta[s]/Zeta[2 s]; m = 110; $MaxExtraPrecision = 470; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*3]]/(2n + 1), {n, 0, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]][[1]]][[1 ;; 105]] (* Jean-François Alcover, Jun 24 2011, after X. Gourdon and P. Sebah, updated Mar 14 2018 *)

PARI

A086033_upto(N=100)={localprec(N+3);digits((PrimeZeta41(3)+1)\.1^N)[^1]} \\ See A086032 for the function PrimeZeta41. - M. F. Hasler, Apr 24 2021

Formula

Zeta_Q(3) = Sum_{p in A002144} 1/p^3 where A002144 = {primes p == 1 (mod 4)};

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(3m)*zeta(3m)/zeta(6m)/(1+8^-m))) [using Gourdon & Sebah, Theorem 11]. - M. F. Hasler, Apr 26 2021

Equals A085541 - 1/2^3 - A085992. - R. J. Mathar, Apr 03 2011