The divisor function for matrices
Tim Browning, Nikita P. Kalinin, Alina Ostafe, Damaris Schindler, Lena Wurzinger
Abstract
We introduce a matrix divisor function τn(T,M), counting factorisations AB=M for n× n integer matrices A,B of height at most T. For a fixed non-singular M, or for the zero matrix M=On, we prove an asymptotic formula for τn(T,M), as T ∞, using lattice point counting. We also prove an essentially sharp uniform upper bound for τn(T,M), for an arbitrary non-singular matrix M.
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