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This is a Quantitative Study study.

October 3, 2025Open Access

Counting 2 2 matrices with fixed determinant and bounded coefficients

Key Points

Findings reveal improved error terms for counting 2x2 matrices with fixed determinant as coefficients grow.
Study shows uniform error terms in both n and X, with X as low as n^(1/2+δ), indicating a threshold in estimates.
Establishes a distinct main term for certain cases involving prime or squared prime determinants, emphasizing field boundaries.
Research enhances understanding of how coefficient bounds affect matrix counting in number theory.

Abstract

Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of 2 2 matrices with fixed non-zero determinant 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. 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.

Counting 2 2 matrices with fixed determinant and bounded coefficients

Key Points

Abstract

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