Hafner-Sarnak-McCurley constant

The Hafner-Sarnak-McCurley constant is a mathematical constant representing the probability that the determinants of two randomly chosen square integer matrices will be relatively prime. The probability depends on the matrix size, "n", in accordance with the formula

:$D(n)=Pi\_\{k=1\}^\{infty\}left\{1-\; [1-Pi\_\{j=1\}^n(1-p\_k^\{-j\})]\; ^2\; ight\},$

where "p_k" is the "k"th prime number. 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

:$Pi\_\{k=2\}^\{infty\}\{zeta(k)\; ^\{-a\_k,$

which converges exponentially; here ζ("k") is the Riemann zeta function.

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 Unpublished manuscript. 1996.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.

External links

*
*