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 n^1/2+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=n^1/2.
Dhanda et al. (Sun,) studied this question.
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