A088540 Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

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

Offset: 1

Author

Benoit Cloitre, Nov 16 2003

Keywords

• cons
• nonn

Comments

An illustration of the Chebyshev effect.

Examples

			1.2923041571286886071...
		

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.

Crossrefs

Cf. A087197, A064533, A088541, A155739.

Programs

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 *)

Formula

Equals (4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).

Equals 4*A087197*A064533/exp(A155739). - R. J. Mathar, Feb 05 2009

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009