A175647 - OEIS

%I #18 Dec 04 2024 09:44:59

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

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

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

%N Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

%C The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...

%C There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.

%H S. Ettahri, O. Ramare, L. Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arxiv:1908.06808 (2019), Section 9.

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.

%F Equals 1/A088539. - _Vaclav Kotesovec_, May 05 2020

%F From _Amiram Eldar_, Sep 27 2020: (Start)

%F Equals Sum_{k>=1} 1/A004613(k)^2.

%F The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)

%e 1.0561821217268161417379307653162198905...

%t digits = 105;

%t LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/  DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];

%t RealDigits[1/(4*LandauRamanujanK/Pi)^2, 10, digits][[1]] (* _Jean-François Alcover_, Jan 12 2021 *)

%Y Cf. A004613, A004614, A013661, A175646.

%K cons,nonn

%O 1,3

%A _R. J. Mathar_, Aug 01 2010

%E More digits from _Vaclav Kotesovec_, Jun 27 2020