A088541 - OEIS

%I #30 Nov 27 2024 16:50:46

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

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

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

%N Decimal expansion of sqrt(Pi)/(2K)*exp(-gamma/2) where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

%C An illustration of the Chebyshev effect.

%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://arxiv.org/abs/2401.00270">A number-theoretic problem concerning pseudo-real Riemann surfaces</a>, arXiv:2401.00270 [math.NT], 2023. See page 6.

%H S. Uchiyama, <a href="http://dx.doi.org/10.1090/S0002-9939-1971-0277494-X">On some products involving primes</a>, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

%F Equals sqrt(Pi)/(2K)*exp(-gamma/2) = lim x-->oo prod(1-1/p) where p runs through the primes p==3 mod 4 and p<=x.

%F Equals A002161*A064533/(2*exp(A155739)). - _Michel Marcus_, Jun 19 2020

%e 0.868927768234323...

%t digits = 104; 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]; Sqrt[Pi]/(2*LandauRamanujanK )*Exp[-EulerGamma/2] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Mar 04 2013, updated Mar 14 2018 *)

%Y Cf. A002161, A064533, A088540, A155739.

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Nov 16 2003