Mesoscopic central limit theorem for general $\beta$-ensembles

We prove that the linear statistics of eigenvalues of $\beta$-log gasses satisfying the one-cut and off-critical assumption with a potential $V \in C^6($\mathbb{R}$)$satisfyacentrallimittheorematallmesoscopicscales$\alpha \in (0; 1)$\mathrm{.}Weprovethisforcompactlysupportedtestfunctions$f \in C^5($\mathbb{R}$)$ using loop equations at all orders along with rigidity estimates.