Hafner-Sarnak-McCurley constant
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The Hafner-Sarnak-McCurley constant is a mathematical constant representing the probability that two randomly chosen matrix determinants will be relatively prime. The probability depends on the matrix size, n, in accordance with the formula
![D(n)=\Pi_{k=1}^{\infty}\left\{1-[1-\Pi_{j=1}^n(1-p_k^{-j})]^2\right\},](../../../math/d/0/7/d0713330124f8206dbdc8bd6ce32b2c4.png)
where pk is the kth prime. The constant is the limit of this expression as n approaches infinity. Its value is roughly 0.3532363719...; Ilan Vardi has given it the alternate expression
which converges exponentially; here ζ is the Riemann zeta function.
[edit] 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 Unpublished manuscript. 1996.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.
[edit] External links
- Weisstein, Eric W. "Hafner-Sarnak-McCurley Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html
