Given two randomly chosen integer matrices,
what is the probability that the corresponding determinants
are relatively prime? Hafner et al. (1993)
showed that
|
(1)
|
where
is the th
prime.
The case is just the probability that two random integers
are relatively prime,
|
(2)
|
(OEIS A059956). No analytic results are known for .
Approximate values for the first few are given by
Vardi (1991) computed the limit
|
(7)
|
(A085849). The speed of convergence is roughly
(Flajolet and Vardi 1996).
See also
Integer Matrix,
Relatively
Prime
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References
Finch, S. R. "Hafner-Sarnak-McCurley Constant." §2.5 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 110-112,
2003.Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical
Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.Hafner,
J. L.; Sarnak, P.; and McCurley, K. "Relatively Prime Values of Polynomials."
In A
Tribute to Emil Grosswald: Number Theory and Related Analysis (Ed. M. Knopp
and M. Seingorn). Providence, RI: Amer. Math. Soc., 1993.Sloane,
N. J. A. Sequences A059956 and A085849 in "The On-Line Encyclopedia of Integer
Sequences."Vardi, I. Computational
Recreations in Mathematica. Redwood City, CA: Addison-Wesley, 1991.Referenced
on Wolfram|Alpha
Hafner-Sarnak-McCurley Constant
Cite this as:
Weisstein, Eric W. "Hafner-Sarnak-McCurley Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html
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