Counting $2\times 2$ integer matrices with a given determinant

Given positive integers $h,N$ satisfying $1⩽h⩽2N^2$, we define $T(h,N)$ to be the number of $2\times 2$ integer matrices with determinant equal to $h$ whose entries lie in $[-N,N]$. Our main result states that for any $\epsilon >0$, one has $$T(h,N)=\frac{16}{\zeta (2)}{N}^2\left(\sum _{d\mathrm{\mid }h}\frac{1}{d}\right)+O_{\epsilon }(N^{\epsilon }(N+h))\mathrm{.}$$ This quantitatively improves upon recent work of Afifurrahman and Ganguly–Guria, and delivers square-root cancellation estimates when $h\le N$. We further show that when $h$ is large, the error term is of approximately the correct order.