OFFSET
0,1
COMMENTS
An illustration of the Chebyshev effect.
REFERENCES
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100
LINKS
Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 6.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.
FORMULA
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.
EXAMPLE
0.868927768234323...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Nov 16 2003
STATUS
approved