Means of algebraic numbers in the unit disk

Abstract

Schur studied limits of the arithmetic means $s_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty} |s_n| \le 1-\sqrt{e}/2.$ We show that $s_n \to 0$, and estimate the rate of convergence by generalizing the Erd\H{o}s-Tur\'an theorem on the distribution of zeros.