Zeros of sections of power series: deterministic and random

Abstract

We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erdös and H. Fried) of the so called Szegö's power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series.