A085996 Decimal expansion of the prime zeta modulo function at 7 for primes of the form 4k+3.
0, 0, 0, 4, 5, 8, 5, 1, 4, 4, 0, 7, 5, 3, 3, 7, 9, 7, 2, 6, 6, 8, 7, 3, 1, 1, 2, 1, 4, 7, 2, 8, 2, 2, 1, 5, 1, 5, 3, 3, 6, 2, 7, 2, 2, 1, 3, 5, 7, 4, 4, 4, 6, 1, 4, 5, 0, 2, 7, 9, 2, 6, 4, 7, 2, 3, 9, 7, 3, 2, 9, 5, 0, 1, 1, 5, 1, 2, 7, 7, 2, 8, 9, 8, 9, 9, 2, 7, 1, 8, 0, 7, 7, 6, 4, 5, 3, 9, 2, 5, 8, 9, 3, 5, 3
Offset: 0
Examples
0.0004585144075337972668731121472822151533627221357444614502792647239732950115...
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- P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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, value P(m=4, n=3, s=7), page 21.
- OEIS index to entries related to the (prime) zeta function.
Crossrefs
Programs
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Mathematica
b[x_] = (1 - 2^(-x))*(Zeta[x]/DirichletBeta[x]); $MaxExtraPrecision = 275; m = 40; Join[{0, 0, 0}, RealDigits[(1/2)* NSum[MoebiusMu[2n + 1]* Log[b[(2n + 1)*7]]/(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
A085996_upto(N=100)={localprec(N+3); digits((PrimeZeta43(7)+1)\.1^N)[^1]} \\ see A085991 for the PrimeZeta43 function. - M. F. Hasler, Apr 25 2021
Formula
Zeta_R(7) = Sum_{primes p == 3 mod 4} 1/p^7
= (1/2)*Sum_{n=0..inf} mobius(2*n+1)*log(b((2*n+1)*7))/(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