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

0,2

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P. Flajolet and I. Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.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

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.

Example

0.14843365646700782822586507749... = 1/3^2 + 1/7^2 + 1/11^2 + 1/19^2 + 1/23^2 + ...

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 *)

Prog

(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

Crossrefs

Cf. A086032 (analog for primes 4k+1), A085548 (PrimeZeta(2)), A002145.

Cf. A085992 .. A085998 (Zeta_R(3..9)).

Sequence in context: A365944 A092511 A045816 * A122110 A082632 A296481

Adjacent sequences: A085988 A085989 A085990 * A085992 A085993 A085994

Keyword

cons,nonn

Author

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

Status

approved