Let $A \in M_n(\mathbb Z)$ and $\|A\| = \max |a_{ij}|$. Denote $$ S(r) = \sum_{\substack{\|A\| \leq r \\\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$ - the sum over all matrices $A \in M_n(\mathbb Z)$ with $\|A\| \leq r$ and $\det{A} \neq 0$.
I'm interested about the asymptotic behavior of the function $S(r)$:
Suggestions on special cases when $n = 2,3$ are also appreciated.
RV above links to a paper of Duke, Rudnick and Sarnak where they establish the following (see Example 1.6 and Theorem 1.10):
The number of $n \times n$ matrices with $||A|| \leq r$ and $\det A = k$ is $$c_n r^{n^2-n} \sum_{d_1 d_2 \cdots d_n=k} \frac{1}{d_2 d_3^2 \cdots d_n^{n-1}} + O(r^{n^2-n-1/n + \epsilon})$$ where $$c_n = \frac{\pi^{n^2-n}}{\Gamma\left(\frac{n}{2}\right) \Gamma\left(\frac{n^2-n+2}{2}\right) \zeta(2) \zeta(3) \cdots \zeta(n)}.$$
Taking $k=1$, we get the lower bound $$S(r) > (c_n-o(1)) r^{n^2-n}.$$ I didn't read the paper carefully enough to figure out how uniform the $O(\ )$ bound is. If we ignored the $O(\ )$ term altogether, we'd get $$S(r) \approx c_n r^{n^2-n} \sum_k \sum_{d_1 d_2 \cdots d_n=k} \frac{1}{k d_2 d_3^2 \cdots d_n^{n-1}} = c_n r^{n^2-n} \sum_{d_1 d_2 \cdots d_n} \frac{1}{d_1 d_2^2 d_3^3 \cdots d_n^{n}}$$ $$ = c_n r^{n^2-n} \sum_{d_1} \frac{1}{d_1} \sum_{d_2} \frac{1}{d_2^2} \sum_{d_3} \frac{1}{d_3^3} \cdots \sum_{d_n} \frac{1}{d_n^n} = c_n r^{n^2-n} \left( \sum_d \frac{1}{d} \right) \zeta(2) \zeta(3) \cdots \zeta(n)$$ Hmm, the sum diverges. That isn't good.
Presumably, if we looked at the proof and the error bounds a little more carefully, we'd wind up cutting off the sum at some upper bound: $c_n r^{n^2-n} \zeta(2) \zeta(3) \cdots \zeta(n) \sum_{d \leq D} 1/d \approx c_n r^{n^2-n} \zeta(2) \zeta(3) \cdots \zeta(n) \log D$. I'm not going to work hard enough to figure out what $D$ is, but surely $n! r^n$ is large enough, as there are no terms with determinant larger than that. On the other hand, I would guess that the leading term in the DRS bound dominates when $k \leq r^{1/(100 n)}$. Both of these are powers of $r$, so both of them have logarithm of the form $c \log r$.
So I would guess that the asymptotics of $S(r)$ are like $$c r^{n^2-n} \log r$$ for some constant $c$ (dependent on $n$).
This is only a partial answer when $n=2$.
When $n=2$, it is easier to visualize.
First, use partial summation to $S(r)$, then we have
$$ S(r)=\int_{1-}^{\infty} \frac{1}{t} dA_t = \frac{A_t}{t}\mid_{1-}^{\infty}+\int_{1-}^{\infty}\frac{1}{t^2}A_tdt,$$ where $$ A_t=\sum_{\substack{{||A||\leq r} \\\ {|\textrm{det}(A)|\leq t}}}1. $$
Interpreting the $|\textrm{det}(A)|$ as the area of parallelopiped spanned by columns of $A$, we find that $$A_t \leq 4\sqrt{2} r^3 t+O(r^3). $$
Hence, we have $$ S(r)\leq 8\sqrt{2} r^3 \log r + O(r^3). $$
Therefore $S(r)/r^4 \rightarrow 0$.
For $n=2$, the growth of $S(r)$ is actually significantly slower than i707107's bounds would suggest. I will show that $A_t = O(t r^2)$, not $t r^3$, and hence $S(r) = O(r^2 \log r)$. I can also establish the lower bound $S(r) > c r^2$, so $2$ is the correct exponent.
Let $A(r,t)$ be the number of $2 \times 2$ matrices with $||A|| \leq r$ and $|\det A| \leq t$, $\det A \neq 0$. We will show $A(r,t) \leq C r^2 t$ and $A(r,1) \geq c r^2$ for some constants $c$ and $C$.
The upper bound: We will count the number of matrices $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$ with $0 < |\det A| \leq t$, $||A|| \leq r$ and $|d| = \max(|a|,|b|,|c|,|d|)$. Multiplying by $4$ then gives an upper bound on $A(r,t)$. We break up our count according to $GCD(c,d)$.
First, let's count the terms with $GCD(c,d)=1$. Fix a particular $(c,d)$. For each $u$ between $-t$ and $t$, the equation $ad-bc=u$ determines $b \bmod d$. Combined with the condition $|b| \leq |d|$, there are only two values for $b$, given a fixed $(c,d)$ and a fixed value for $\det \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$. And, $(b,c,d,u)$ determines $a$. There are $O(r^2)$ choices for $(c,d)$, and $O(t)$ choices for $u$, so such terms contribute $O(t r^2)$ to our count.
Now, let's do the case where $GCD(c,d) = g$, say $c = g c'$ and $d = g d'$. The equation $ad-bc=u$ is only solvable at all when $g|u$, which is for $O(t/g)$ values of $u$. In that case, $b$ is determined modulo $d'$. Combined with the inequality $|b|\leq d$, this gives $O(d/d') = O(g)$ choices for $b$. So, for each $(c,d)$, there are $O(g) O(t/g) = O(t)$ ways to complete it to a $(a,b,c,d,u)$ quintuple. The number of $(c,d)$ pairs with $GCD$ equal to $g$ is $O(r^2/g^2)$.
So our bound is $O \left( \sum_g t r^2/g^2 \right) = O(t r^2)$.
The lower bound There are $(6/\pi^2 - o(1)) r^2$ pairs $(c,d)$ with $GCD(c,d)=1$ and $|c|, |d| \leq r^2$. For each of these, we can find $(a,b)$ with $|a|$, $|ab| \leq \max(|c|, |d|)$ and $ad-bc=1$. So there are $(6/\pi^2 - o(1)) r^2$ matrices contributing to $A(r,1)$. (It is easy to improve this constant, but probably not worthwhile.)
