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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.16890 |
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| _version_ | 1866908925499015168 |
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| author | Dhanda, Kavita Haynes, Alan Prasala, Silmi |
| author_facet | Dhanda, Kavita Haynes, Alan Prasala, Silmi |
| contents | Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, 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 $X\rightarrow\infty$. 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 $n^{1/2+δ}$. 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 $X=n^{1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $2\times 2$ matrices with fixed determinant and bounded coefficients Dhanda, Kavita Haynes, Alan Prasala, Silmi Number Theory 11D04, 11D45, 11N45 Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, 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 $X\rightarrow\infty$. 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 $n^{1/2+δ}$. 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 $X=n^{1/2}$. |
| title | Counting $2\times 2$ matrices with fixed determinant and bounded coefficients |
| topic | Number Theory 11D04, 11D45, 11N45 |
| url | https://arxiv.org/abs/2509.16890 |