The Hafner-Sarnak-McCurley constant is the limit as 
of the limiting probability 
that two 
integer matrices
have relatively prime determinants.
For each positive integer 
, choose their entries independently and uniformly from 1,
2, ..., 
,
then take 
. Hafner et al. (1993, p. 438) showed
that
![D(n)=product_(k=1)^infty{1-[1-product_(j=1)^n(1-p_k^(-j))]^2},](/images/equations/Hafner-Sarnak-McCurleyConstant/NumberedEquation1.svg) |
(1)
|
where 
is the 
th
prime.
The case 
is just the probability
that two random integers are relatively
prime,
 |
(2)
|
(OEIS A059956). Approximate values for the first few 
are given by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Vardi (1991) computed the limit
 |
(7)
|
(OEIS A085849). Flajolet and Vardi (1996, pp. 5-6) give a zeta function product for this constant whose
convergence rate is roughly 
after 
factors.
