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\},

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

\Pi_{k=2}^{\infty}{\zeta(k) ^{-a_k}},

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 Unpublished manuscript. 1996.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

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