OFFSET
1,2
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 5.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.
FORMULA
(4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).
EXAMPLE
1.2923041571286886071...
MATHEMATICA
digits = 105; 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]; 4/Sqrt[Pi]*Exp[-EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Jun 04 2014, updated Mar 14 2018 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Nov 16 2003
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved