I'm interested in this function

$
h(m,n) = sgn(\lim_{\substack{\epsilon \to 0 \\ w \to \infty}}{\sum_{x_1=1}^{w}...\sum_{x_u=1}^{w}\dfrac{\epsilon^2}{\epsilon^2 + p^2(m,n,x_1,...,x_u)}})
$

how can we simplify it?