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Source. Theorem I, p. 106, proof in sections 10--14, pp. 112--118, of P. Erdős and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), no. 1, 105--119, DOI 10.2307/1969500, the edition named on the source card.

Statement

Notation (p. 105, (1.3), "as throughout the present paper"): for a polynomial with coefficients $a_0,\ldots,a_n$,

$$P=\frac{\lvert a_0\rvert+\cdots+\lvert a_n\rvert}{\sqrt{\lvert a_0a_n\rvert}}.$$

Theorem I (p. 106, quoted). "If the roots of the polynomial $f(z)=a_0+a_1z+\cdots+a_nz^n$ are denoted by $z_\nu=r_\nu e^{i\varphi_\nu}$, $\nu=1,2,\cdots,n$ then for every $0\le\alpha<\beta\le2\pi$ we have

$$\Bigl\lvert\sideset{}{_\nu}\sum_{\alpha\le\varphi_\nu\le\beta}1-\frac{\beta-\alpha}{2\pi}n\Bigr\rvert <16\sqrt{n\log\frac{\lvert a_0\rvert+\cdots+\lvert a_n\rvert}{\sqrt{\lvert a_0a_n\rvert}}} =16\sqrt{n\log P}."$$

(The display is the paper's (3.3); (3.1) and (3.2) are the two displayed formulas inside the quotation.)

So the count, with multiplicity, of the roots whose argument lies in the closed interval $[\alpha,\beta]$ differs from $(\beta-\alpha)n/2\pi$ by less than $16\sqrt{n\log P}$, uniformly over all such intervals. The theorem names no further hypothesis; the quantity $P$ is defined only when $a_0a_n\ne0$, so the degree is exactly $n$ and no root is $0$, and every root then has an argument $\varphi_\nu$ in $[0,2\pi)$. The proof (p. 118) uses $P\ge2$.

Consequence printed with it (p. 107, (3.4)--(3.5)). If $n^{-\lambda}\le\lvert a_\nu\rvert\le n^\lambda$ for $\nu=0,1,\ldots,n$, then $P\le(n+1)n^{2\lambda}<(n+1)^{2\lambda+1}$, and the discrepancy is less than $16\sqrt{2\lambda+1}\,\sqrt{n\log(n+1)}$.

Read depth. Claims checked: the definition (1.3), the statement (3.3) and the consequence (3.4)--(3.5) were read clause by clause on the page images of pp. 105--107. The proof on pp. 112--118 was read for its structure, not checked line by line. Nothing here is independently reviewed.

Proof pointer

Sections 10--14, pp. 112--118, written here in outline. With $g(z)=\prod_\nu(z-e^{i\varphi_\nu})$, the polynomial whose roots are those of $f$ moved radially onto the unit circle, a remark the paper credits to Schur gives $\lvert g(z)\rvert\le P$ on $\lvert z\rvert=1$ ((10.3), p. 112). Applying an upper bound for the number of roots of $g$ in an arc twice, to the two complementary arcs, gives the lower bound too (p. 113), so it suffices to prove the upper bound (10.6) with constant $8$. That bound comes from an extremal problem: among polynomials of degree $n$ with leading coefficient of modulus $1$, all roots on the unit circle and exactly $K+2l+1$ roots on the arc, where $K=[\delta n/2\pi]$ ((11.1)--(11.2), p. 113), the minimal maximum modulus is attained by a polynomial that, by the Lemma of p. 114 and a theorem of Turán on the spacing of roots near a maximum point (p. 115), has a root of multiplicity $l$; a weighted $L^2$ minimum computed by a theorem of Szegő (pp. 115--117) then bounds $l$ by $2\sqrt{(n+1)\log P}$, and (14.8) on p. 118 finishes the count.

Dependencies

A remark of Schur (p. 112), a theorem of Turán on the roots near a maximum point on the unit circle (p. 115, cited from Szeged Acta 11 (1946), 106--113), and a theorem of Szegő on extremal integrals (p. 115, cited from his Orthogonal Polynomials, p. 282, Theorem 11.1.2).

Bears on

• Problem 990: the problem asks whether the discrepancy of the root arguments over intervals is $\ll(n\log M)^{1/2}$ with $n$ the number of nonzero coefficients and $M$ the paper's $P$. Theorem I proves a bound of that shape with the degree in place of the number of nonzero coefficients, with the constant $16$, for closed intervals $[\alpha,\beta]\subseteq[0,2\pi]$ and polynomials with $a_0a_n\ne0$. The paper does not consider the number of nonzero coefficients.