Abstract: Erdos and Turan prove a classical inequality for complex polynomials, which says if the polynomial attains small value on the unit circle after normalization, then all zeros will cluster around the unit circle and moreover become equidistributed in angles. The optimal constant remained the only component that is not sharp for this inequality. We will explain how tools in potential theory and energy minimization enter this question, and how they help us in characterizing the extremal distribution of zeros and proving the optimal constant. This is a joint work with Ruiwen Shu.
