\displaystyle J=\sum_{1\leq m,n\leq N;\\m\neq n}\int e^{-\pi(|nv-x|^2+|mv-x|^2)}e^{i(\beta_n-\beta_m)} =(?)\sum_{n\neq m} \int e^{-\pi(|nv-x|^2+|mv-x|^2)}\to 0 