Stacky Heights, Entropy, and Zeta Functions of Weighted Hypersurfaces over Finite Fields

Let \(X\subset \mathbb P^n_{\mathbf w}\) be a weighted hypersurface over \(\mathbb F_q\), with associated stack \(\mathfrak X\). For \(\mathbf x\in X(\mathbb F_{q^r})\), let \(g_{\mathbf x}(r)=\gcd(k_{S(\mathbf x)},q^r-1)\), where \(k_S=\gcd(w_i:i\in S)\), and define \(Θ_r(s)=\sum_{\mathbf x}g_{\mat