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Hafner-Sarnak-McCurley constant
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Numbers
$\omega$
0.353236371854995984543516550432682011280164778566690446416085942814238325002669003483672078334335498967
Definition
The Hafner-Sarnak-McCurley constant $\omega$ is the limit of the probabilities $D(n)$ as $n \to \infty$ that the determinants of two random $n\times n$ matrices with integral coefficients have coprime determinants.
Formulas
(1)
$\omega = \prod_p \big(1- \big(1 - \prod_{j=1}^\infty (1-p^{-j})\big)^2 \big)$, where the outer product ranges over all primes $p$.
(2)
$D(n) = \prod_p \big( 1 - \big( 1 - \prod_{j=1}^n (1 - p^{-j}) \big)^2 \big)$.
References
[1]
Flajolet, P. and Vardi, I., "Zeta Function Expansions of Classical Constants", Unpublished manuscript (1996). http://algo.inria.fr/flajolet/Publications/landau.ps
[2]
L. Hafner, P. Sarnak and K. McCurley, "Relatively prime values of polynomials", In A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics (1993), M. Knopp and M. Sheigorn, Editors, vol. 143.