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

TL;DR

This work derives a uniform asymptotic formula for the number of 2x2 integer matrices with determinant $\Delta$ and entries bounded by $H$, establishing $\#{\mathcal D}_2(H,\Delta)=\dfrac{96}{\pi^2}\dfrac{\sigma(|\Delta|)}{|\Delta|}H^2+O\bigl(H^{o(1)}\max(H^{5/3},|\Delta|)\bigr)$ for $\Delta\neq 0$, uniformly in a broad range of $|\Delta|$. The method reduces counting to solutions of $ad- bc=\Delta$ by partitioning ranges of $a$ and $c$, and applies results on modular hyperbolas (uv $\equiv K\pmod{q}$) due to Ustinov, complemented by summation identities for gcd- and divisor-related functions. It also treats the $\Delta=0$ case, showing $\#{\mathcal D}_2(H,0)=\dfrac{96}{\pi^2}H^2\log H+O(H^2)$, and connects the $\Delta$-case counts to the second moment of the restricted divisor function $\tau_N$. The findings advance uniform arithmetic statistics for integer matrices and yield explicit divisibility-sum expressions, with potential applications to moments of divisor-type functions and lattice-point counting in modular regions.

Abstract

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

Theorems & Definitions (15)

• Theorem 1.1
• Proposition 1.2
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Corollary 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7