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Source. Theorem II, p. 109, proof omitted on p. 110, 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

Setting (p. 108, (6.1) and (6.4)). The power series $f(z)=1+a_1z+\cdots+a_nz^n+\cdots$ is regular for $\lvert z\rvert<1$ with the unit circle as its circle of convergence, and its sections are $s_n(z)=\sum_{j=0}^{n}a_jz^j$ with $a_0=1$.

Theorem II (p. 109, quoted). "If for the coefficients of the power-series (6.1) we have

$$\nu^{-\lambda}\le\lvert a_\nu\rvert\le\nu^\lambda,\qquad\nu=1,2,\cdots \tag{7.1}$$

then there is a $A_{12}=A_{12}(\lambda)$ such that for the roots $z_1,z_2,\cdots z_n$ of the section $s_n(z)$ we have for any $0\le\alpha<\beta\le2\pi$

$$\Bigl\lvert\sideset{}{_j}\sum_{\substack{\alpha\le\operatorname{arc}z\text{ [sic] }\le\beta\\ 1-(1/\sqrt n)\le\lvert z_j\rvert\le1+(1/\sqrt n)}}1-\frac{\beta-\alpha}{2\pi}n\Bigr\rvert <A_{12}(\lambda)\sqrt{n\log n}."$$

So the roots of $s_n$ that lie in the closed annulus $1-1/\sqrt n\le\lvert z\rvert\le1+1/\sqrt n$ and in the closed sector $\alpha\le\arg z\le\beta$ number $(\beta-\alpha)n/2\pi$ up to an error less than $A_{12}(\lambda)\sqrt{n\log n}$, with $A_{12}$ depending only on $\lambda$. In the print the first condition under the sum reads $\operatorname{arc}z$, without the index $j$. The statement gives no range for $\lambda$ or for $n$; at $n=1$ the right side is $0$. The hypothesis (7.1) already forces the radius of convergence to be $1$ (an observation here, not a statement of the paper).

Read depth. Claims checked: the statement and the setting (6.1), (6.4) were read clause by clause on the page images of pp. 108--109. The paper gives no proof to check.

Proof pointer

None in the paper: "The proof goes along the same lines as in §6 so we can omit the details" (p. 110). Section 6 (pp. 108--109) applies Theorem I to the sections of the series, bounding $P$ through upper and lower bounds on the coefficients, and then counts the roots outside an annulus about the unit circle through the product of the moduli of the roots.

Dependencies

Theorem I (p. 106) and the method of section 6 (pp. 108--109).

Bears on

No Erdős problem page cites this result.