Hafner-Sarnak-McCurley constant

The Hafner – Sarnak – McCurley constant is a mathematical constant that indicates the probability with which the determinants of two matrices are coprime to one another .

definition

Let be two square, integer matrices. Then the probability that the two determinants are coprime to one another is given by the function ${\displaystyle N,M\in \mathbb {Z} ^{n\times n}}$ ${\displaystyle n\times n}$

${\displaystyle D(n)=\prod _{k=1}^{\infty }(1-(1-\prod _{j=1}^{n}1-p_{k}^{-j})^{2})}$

described. The n-prime number denotes . ${\displaystyle p_{n}}$

Graph of the D (n) function

In particular, for two matrices the probability of coprime is: ${\displaystyle 1\times 1}$

${\displaystyle D(1)={6 \over \pi ^{2}}=0{,}607927}$. (OEIS A059956 )

Other values

The exact function values ​​for were not determined analytically. The approximate values ​​are: ${\displaystyle n\geq 2}$

n	D (n)
2	0.453103
3	0.397276
4th	0.373913
5	0.363321

limit

Vardi (1991) determined the limit value for the function ${\displaystyle D(n)}$

${\displaystyle \omega =\lim _{n\to \infty }D(n)=0{,}35323637185}$( A085849 )

proved with an approximation speed of . ${\displaystyle \sim 0{,}57^{n}}$

literature

• Finch, SR (2003), "§2.5 Hafner-Sarnak-McCurley Constant", Mathematical Constants, Cambridge, England: Cambridge University Press, pp. 110-112, ISBN 0-521-81805-2
• Hafner, JL; Sarnak, P. & McCurley, K. (1993), "Relatively Prime Values ​​of Polynomials", in Knopp, M. & Seingorn, M. (eds.), A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., ISBN 0-8218-5155-1
• Vardi, I. (1991), Computational Recreations in Mathematica, Redwood City, CA: Addison-Wesley, ISBN 0-201-52989-0

Individual evidence

1. ↑ Hafner, Sarnak, McCurley, op. Cit.
2. ^ Eric W. Weisstein: Hafner-Sarnak-McCurley Constant. Retrieved June 16, 2019 .