  Schur studied limits of the arithmetic means $A_n$ 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
$\limsup_{n\to\infty} |A_n| \le 1-\sqrt{e}/2.$ We show that $A_n \to 0$, and
estimate the rate of convergence by generalizing the Erd\H{o}s-Tur\'an theorem
on the distribution of zeros. As an application, we show that integer
polynomials have some unexpected restrictions of growth on the unit disk.
  Schur also studied problems on means of algebraic numbers on the real line.
When all conjugate algebraic numbers are positive, the problem of finding the
sharp lower bound for $\liminf_{n\to\infty} A_n$ was developed further by
Siegel and others. We provide a solution of this problem for algebraic numbers
equidistributed in subsets of the real line.
  Potential theoretic methods allow us to consider distribution of algebraic
numbers in or near general sets in the complex plane. We introduce the
generalized Mahler measure, and use it to characterize asymptotic
equidistribution of algebraic numbers in arbitrary compact sets of capacity
one. The quantitative aspects of this equidistribution are also analyzed in
terms of the generalized Mahler measure.
