Sum rules & Tauberian theorems at finite temperature

A bstract We study CFTs at finite temperature and derive explicit sum rules for one-point functions of operators by imposing the KMS condition and we explicitly estimate one-point functions for light operators. Turning to heavy operators we employ Tauberian theorems and compute the asymptotic OPE density for heavy operators, from which we extract the leading terms of the OPE coefficients associated with heavy operators. Furthermore, we approximate and establish bounds for the two-point functions.