A086034 Decimal expansion of the prime zeta modulo function at 4 for primes of the form 4k+1.

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

Offset: 0

Author

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

Keywords

• cons
• nonn

Examples

			0.0016495841540292915989967613136388518274879099438347321478115258388...
		

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, value P(m=4, n=1, s=4), page 21.
• OEIS index to entries related to the (prime) zeta function.

Crossrefs

Cf. A085993 (same for primes 4k+3), A343624 (for primes 3k+1), A343614 (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 = 120; $MaxExtraPrecision = 680; Join[{0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]*Log[a[(2n + 1)*4]]/(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

A086034_upto(N=100)={localprec(N+3); digits((PrimeZeta41(4)+1)\.1^N)[^1]} \\ see A086032 for the PrimeZeta41 function. - M. F. Hasler, Apr 26 2021

Formula

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

= Sum_{odd m > 0} mu(m)/2m * log(DirichletBeta(4m)*zeta(4m)/zeta(8m)/(1+2^(-4m)))[using Gourdon & Sebah, Theorem 11] - M. F. Hasler, Apr 26 2021.

Extensions

Edited by M. F. Hasler, Apr 26 2021