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arXiv · 2509.20259

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

Jonathan Chapman·Akshat Mudgal
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Abstract

Given positive integers $h, N$ satisfying $1 \leqslant h \leqslant 2N^2$, 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 $\varepsilon >0$, one has \[ T(h,N) = \frac{16}{\zeta(2)} N^2 \bigg( \sum_{d |h} \frac{1}{d} \bigg) + O_{\varepsilon}(N^{\varepsilon} (N+ h)).\] This quantitatively improves upon recent work of Afifurrahman and Ganguly--Guria, and delivers square-root cancellation estimates when $h \leq N$. We further show that when $h$ is large, the error term is of approximately the correct order.

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Jonathan Chapman, Akshat Mudgal. 2025-09-24. Counting $2 \times 2$ integer matrices with a given determinant. https://arxiv.org/abs/2509.20259

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