{"paper":{"title":"Counting $2 \\times 2$ integer matrices with a given determinant","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Akshat Mudgal, Jonathan Chapman","submitted_at":"2025-09-24T15:45:45Z","abstract_excerpt":"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 approxim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.20259","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.20259/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}