Counting 2 × 2 integer matrices with a given determinant

2025/09/24 by Jonathan Chapman, Chapman, Jonathan, Akshat Mudgal +1
Computer Science · Engineering · Mathematics · #11D09 #11D45 #11N37 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2509.20259

openalex publication_date 2025/09/24 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28

Abstract

Given positive integers h, N satisfying 1 \leqslant h \leqslant 2N², we define T(h,N) to be the number of 2× 2 integer matrices with determinant equal to h whose entries lie in [-N,N]. Our main result states that for any ε >0, one has T(h,N) = (16)/(ζ(2)) N² ( ∑_{d |h} (1)/(d) ) + O_ε(N^ε (N+ h)). This quantitatively improves upon recent work of Afifurrahman and Ganguly--Guria, and delivers square-root cancellation estimates when h ≤ N. We further show that when h is large, the error term is of approximately the correct order.

Citations

• On commuting pairs in arbitrary sets of 2x2 matrices
• Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian
• A uniform formula on the number of integer matrices with given determinant and height
• Hyperbolic lattice point counting in unbounded rank
• On the asymmetric additive energy of polynomials
• Counting lattice points in norm balls on higher rank simple Lie groups
• Moments of random multiplicative functions and truncated characteristic polynomials
• High moments of the Riemann zeta-function
• Khintchine’s problem in metric Diophantine approximation

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• On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant