Counting 2 times 2 2 × 2 $2\times 2$ matrices with fixed determinant and bounded coefficients

Abstract Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of 2 times 2 2 × 2 $2\times 2$ matrices with fixed non-zero determinant n element of double struck upper N n ∈ N $n\in\mathbb{N}$ , 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 upper X right arrow normal infinity X → ∞ $X\rightarrow\infty$ . 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 Superscript 1 divided by 2 plus delta n 1 / 2 + δ $n^{1/2+\delta}$ . To complement this, we also demonstrate that the exponent 1 divided by 2 plus delta 1 / 2 + δ $1/2+\delta$ 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 upper X equals n Superscript 1 divided by 2 X = n 1 / 2 $X=n^{1/2}$ .