Means of algebraic numbers in the unit disk

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 ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}\mathrm{\mid }{s}_n\mathrm{\mid }\le 1-\sqrt{e}\mathrm{/}2.$ 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.