Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Let m,n≥2m,n≥2, m≤nm≤n. It is well-known that the number of (two-dimensional) threshold functions on an m×nm×n rectangular grid is t(m,n)=6π2(mn)2+O(m2nlogn)+O(mn2loglogn)=6π2(mn)2+O(mn2logm). We improve the error term by showing that t(m,n)=6π2(mn)2+O(mn2).