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Source. Theorem 4, p. 1173, of P. Erdős, "Some remarks on polynomials," Bull. Amer. Math. Soc. 53 (1947), 1169-1176. Pages are the journal's own, as on the source card.

Setting

Page 1173, following Schur (Math. Z. 1 (1918), Theorem XIII, pp. 397-398). Let $a_0$ be a given integer and let $f_n(z)=a_0z^n+\cdots+a_n$ be a polynomial with integer coefficients whose roots $z_1,\ldots,z_n$ either all have absolute value $1$ and are distinct, or all lie in the interior of the unit circle, in which case multiple roots are permitted. Schur proved

$$\limsup\frac{z_1+z_2+\cdots+z_n}{n}\le1-\frac{e^{1/2}}{2}, \tag{6}$$

as printed, conjectured that the limit is $0$, and remarked that for $a_0=1$ this follows from Kronecker's theorem, since then all the $z_i$ are roots of unity.

Statement

Theorem 4 (p. 1173). For the $z_i$ as above,

$$\lim\frac{z_1+z_2+\cdots+z_n}{n}=0 .$$

The limit is taken as $n\to\infty$; the proof notes that for each $n$ there are only finitely many such polynomials.

Read depth. Claims checked: the statement and the setting were read on the print. The proof (pp. 1173-1174) was read but not checked step by step.

Proof pointer

Pages 1173-1174. Polynomials with all roots inside the circle are reduced to polynomials with distinct roots on the circle having the same sum of roots, following Schur (p. 397). For roots on the circle the discriminant $D=a_0^{2n-2}\prod_{i<j}(z_i-z_j)^2$ is an integer at least $1$, (7), while a result of Pólya bounds $\prod_{i<j}\lvert z_i-z_j\rvert$ by $n^n$, (8). It suffices to show the roots are uniformly distributed on the circle. If they are not, a result of Fekete (Ann. of Math. 41 (1940), pp. 165-166) gives a point of the circle at which the product of distances to the roots is exponentially large, (10); replacing roots repeatedly, about $c_2n$ times, yields points on the circle whose product of mutual distances exceeds $n^n$, contradicting (8).