I wonder if there are good estimate for the number of elements in $\text{SL}(d,\mathbb Z)$ with $2$-norm bounded by $T$ as $T \to \infty$, namely
$$\#\{g\in SL(d,\mathbb Z):\|g\|_2:=\sqrt{ \sum_{ij}g_{ij}^2} \le T\}.$$
It would be great if there is a sharp estimate for this problem. Any proof (even for the case $d=2$) of reference will be appreciated!