Counting 2× 2 matrices with fixed determinant and bounded coefficients

Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of 2× 2 matrices with fixed non-zero determinant n, and with coefficients bounded in absolute value by X. In this paper we present a new proof of this result, which also gives an improved error term as X→∞. Similar to Afifurrahman's result, our error term is uniform in both n and X, and our estimates are significant for X as small as n1/2+δ. To complement this, we also demonstrate that the exponent $1/2+δ$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when n is either a prime or the square of a prime, and when X=n1/2.