## High-Energy Conformal Bootstrap and Tauberian Theory

We analyze modular invariance drawing inspiration from tauberian

theorems. Given a modular invariant partition function with a positive

spectral density, we derive lower and upper bounds on the number of

operators within a given energy interval. They are most revealing at high

energies. In this limit we rigorously derive the Cardy formula for the

microcanonical entropy together with optimal error estimates for various

widths of the averaging energy shell. Finally, we identify a new universal