A085991 Decimal expansion of the prime zeta modulo function at 2 for primes of the form 4k+3.
1, 4, 8, 4, 3, 3, 6, 5, 6, 4, 6, 7, 0, 0, 7, 8, 2, 8, 2, 2, 5, 8, 6, 5, 0, 7, 7, 4, 9, 0, 7, 1, 1, 3, 7, 1, 8, 8, 7, 5, 5, 5, 8, 4, 1, 7, 4, 4, 8, 0, 6, 8, 8, 9, 4, 4, 2, 5, 0, 7, 5, 0, 8, 0, 5, 5, 2, 9, 8, 2, 0, 0, 3, 1, 9, 7, 6, 8, 2, 2, 9, 3, 0, 6, 4, 3, 0, 9, 8, 6, 8, 5, 0, 6, 7, 2, 4, 6, 9, 0, 9, 3, 5, 0, 7
Offset: 0
Examples
0.14843365646700782822586507749... = 1/3^2 + 1/7^2 + 1/11^2 + 1/19^2 + 1/23^2 + ...
Links
- Jean-François Alcover, Table of n, a(n) for n = 0..999
- P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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, section 3.2 constant P(m=4,n=3,s=2).
Crossrefs
Programs
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Mathematica
digits = 1000; nmax0 = 500; dnmax = 10; Clear[PrimeZeta43]; PrimeZeta43[s_, nmax_] := PrimeZeta43[s, nmax] = (1/2) Sum[(MoebiusMu[2n + 1] ((4n + 2) Log[2] + Log[((-1 + 2^(4n + 2)) Zeta[4n + 2])/(Zeta[4 n + 2, 1/4] - Zeta[4n + 2, 3/4])]))/(2n + 1), {n, 0, nmax}] // N[#, digits+5]&; PrimeZeta43[2, nmax = nmax0]; PrimeZeta43[2, nmax += dnmax]; While[Abs[PrimeZeta43[2, nmax] - PrimeZeta43[2, nmax - dnmax]] > 10^-(digits+5), Print["nmax = ", nmax]; nmax += dnmax]; PrimeZeta43[2] = PrimeZeta43[2, nmax]; RealDigits[PrimeZeta43[2], 10, digits][[1]] (* Jean-François Alcover, Jun 21 2011, updated May 06 2021 *) -
PARI
PrimeZeta43(s)={suminf(n=0, my(t=s+s*n*2); moebius(n*2+1)*log(zeta(t)/(zetahurwitz(t, 1/4)-zetahurwitz(t, 3/4))*(4^t-2^t))/(n*2+1))/2} A085991_upto(N=100)={localprec(N+3); digits((PrimeZeta43(2)+1)\.1^N)[^1]} \\ M. F. Hasler, Apr 25 2021
Formula
Zeta_R(2) = Sum_{primes p == 3 (mod 4)} 1/p^2
= (1/2)*Sum_{n>=0} mobius(2*n+1)*log(b((2*n+1)*2))/(2*n+1),
where b(x)=(1-2^(-x))*zeta(x)/L(x) and L(x) is the Dirichlet Beta function.