The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with Constant Curvature

Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $\rho$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_M^m$. That is, $$\sum _{i=0}^{d_m-1}\mathrm{\parallel }{S}_i(x_0){\mathrm{\parallel }}_{h_m}^2\sim m(1+\frac{\rho }{2m})+O(e^{-\frac{(\log m{)}^2}{8}}),$$ where $\{S_0,\mathrm{.}\mathrm{.}\mathrm{.},S_{d_m-1}\}$ is an orthonormal basis for $H^0(M,K_M^m)$ for sufficiently large $m$.