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 ${\mathrm{limsup}}_{n\to \infty }|s_n|\le 1-\sqrt{e}/2.$ We show that $s_n\to 0$, and estimate the rate of convergence by generalizing the Erdős-Turán theorem on the distribution of zeros.