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arXiv · 2608.25078

The divisor function for matrices

Tim Browning·Nikita P. Kalinin·Alina Ostafe·Damaris Schindler·Lena Wurzinger
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Abstract

We introduce a matrix divisor function $\tau_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $\tau_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove an essentially sharp uniform upper bound for $\tau_n(T,M)$, for an arbitrary non-singular matrix $M$.

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Tim Browning, Nikita P. Kalinin, Alina Ostafe, Damaris Schindler, Lena Wurzinger. 2026-08-25. The divisor function for matrices. https://arxiv.org/abs/2608.25078

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