Counting $2 \times 2$ integer matrices with a given determinant
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914576837115904 |
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| author | Chapman, Jonathan Mudgal, Akshat |
| author_facet | Chapman, Jonathan Mudgal, Akshat |
| contents | 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 \[ T(h,N) = \frac{16}{ζ(2)} N^2 \bigg( \sum_{d |h} \frac{1}{d} \bigg) + O_{\varepsilon}(N^{\varepsilon} (N+ h)).\] 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $2 \times 2$ integer matrices with a given determinant Chapman, Jonathan Mudgal, Akshat Number Theory Combinatorics 11D45, 11D09, 11N37 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 \[ T(h,N) = \frac{16}{ζ(2)} N^2 \bigg( \sum_{d |h} \frac{1}{d} \bigg) + O_{\varepsilon}(N^{\varepsilon} (N+ h)).\] 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. |
| title | Counting $2 \times 2$ integer matrices with a given determinant |
| topic | Number Theory Combinatorics 11D45, 11D09, 11N37 |
| url | https://arxiv.org/abs/2509.20259 |