On non-singularity of integer matrices with bounded entries Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.
 A: In the simplest case $n=2$ is is possible to prove that this probability is equals to $$1-\frac{3\log B}{\pi^2B^2}+O\left(\frac{1}{B^2}\right)$$
for large $B$.
To do this let us observe that every singular matrix is born from integer vector $v=(a,b)$ with coprime $a,b<B$ and with positive maximum component, and consists of vectors $iv,jv$ with integers $i,j$ such that $|i|,|j|<\frac{B}{\max(a,b)}$. Because of zero matrix repeats in counting in $O(B^2)$ cases the quantity of singular matrices equals to 
$$\sum_v 4 \phi(\max(a,b))\left[\frac{B}{\max(a,b)} \right ]^2 +O(B^2)= 8\sum_{b=2}^B \phi(b) \left[\frac{B}{b} \right ]^2 + O(B^2).$$
Sum after coefficient is also equal to:
$$S=\sum_{l=0}^B (2l+1) \sum_{k=2}^{[B/l]} \phi(k).$$ 
Let us remember the summation formala and let us write $\sum_{k \leq n} \phi(k)=3n^2/\pi^2+c(n)n\log n$ with $c(n)<C$ all $n\leq B$. After some manipulations wuth integer part we can also write:
$$ \sum_{k=2}^{[B/l]} \phi(k) = \frac{3B^2}{\pi^2l^2}+c\left (\frac{B}{l} \right ) \frac{B}{l} \log \frac{B}{l}.$$
So $S=S_1+S_2$, where
$$ S_1 = \sum_{l=0}^B (2l+1) \frac{3B^2}{\pi^2l^2}=\frac{6}{\pi^2}B^2 \log B+O(B^2),$$
and
$$S_2 = \sum_{l=0}^B (2l+1) c\left (\frac{B}{l} \right ) \frac{B}{l} \log \frac{B}{l} \leq CB\sum_{l\leq B} (\log B - \log l) = O(B^2).$$
Finally, we have that number of nonsingular matrices equals to $(2B-1)^4-8S+O(B^2)$, so the corresponding probability is equal to 
$$1-\frac{8S}{(2B-1)^4}+O\left(\frac{1}{B^2}\right)=1-\frac{3\log B}{\pi^2B^2}+O\left(\frac{1}{B^2}\right).$$
A: If I understand the question correctly, this is a known hard problem if $B$ is fixed, but $n$ is growing. See, for example, this paper, written by some fairly smart people:
On the singularity probability of discrete random matrices
Jean Bourgain, Van Vu, P. M. Wood (sorry, insert citation is failing), JFA 2010.
However, if $n$ is fixed, but $B$ is growing, this is well-understood, see the paper by Yonatan Katznelson (DMJ, 1994, "Singular matrices and a uniform bound for congruence subgroups of $SL(n, \mathbb{Z})$"), his main result is that the probability is of order $\log B/B^n,$ with explicit constants.
