A uniform formula on the number of integer matrices with given determinant and height

TL;DR

This paper derives an asymptotic formula for counting 2x2 integer matrices with a fixed determinant and bounded entries, valid across a wide range of determinants relative to the entry bound.

Contribution

It provides a uniform asymptotic count for integer matrices with specified determinant and entry height, extending previous results to a broader parameter range.

Findings

Asymptotic formula for the count of matrices with fixed determinant and bounded entries

Uniform validity of the formula over a large range of determinants

Enhanced understanding of the distribution of such matrices

Abstract

We obtain an asymptotic formula for the number of integer $2\times 2$ matrices that have determinant $\Delta$ and whose absolute values of the entries are at most $H$. The result holds uniformly for a large range of $\Delta$ with respect to $H$.