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Statement

Setting (abstract, p. 499). For complex numbers $z_1,\dots,z_n$ let $S_j=z_1^j+\cdots+z_n^j$ and

$$R_n=\min_{z_1,\dots,z_n}\ \max_{1\le j\le n}|S_j|,$$

the minimum taken under the condition $\max_{1\le t\le n}|z_t|=1$. The paper notes (p. 499) that the minimum exists by Weierstrass' theorem and that the condition can be replaced by $z_1=1$.

Theorem (p. 500, quoted). "We have"

$$\limsup_{n\to\infty}R_n<1.$$

Consequences in section 3 (pp. 506--507). For a fixed complex $\alpha$ with $|1-\alpha|<1$ and a fixed number $q$, the paper shows that if inequality (22) holds then $\limsup_{n\to\infty}R_n\le\max(|1-\alpha|,q)$ (23). Taking $q=|1-\alpha|$ it derives the sufficient condition (24), $|1-\alpha|\bigl(\frac1{\operatorname{Re}\alpha}-\frac1{|\alpha|}\bigr)>2^{\operatorname{Re}\alpha}$, for $\limsup_{n\to\infty}R_n\le|1-\alpha|$ (25). With $\alpha=(1+i)/5$ it checks (24) and has $|1-\alpha|^2=17/25<(5/6)^2$ (26), and concludes $R_n<5/6$ for all large enough $n$ (p. 507).

Addendum (p. 507). The paper reports that G. Harcos, by computer work based on (22) and (23), found that $\alpha=0.56754+0.54237i$ gives $\limsup_{n\to\infty}R_n<0.69368$; the introduction (p. 500) states this as $R_n<0.694$ for large $n$. The value is a reported computation, not a computation carried out in the paper. Harcos also observed that the identity (14) can be derived from the inverse Newton--Girard formulas.

Source. A. Biró, An upper estimate in Turán's pure power sum problem, Indag. Math. (N.S.) 11 (2000), no. 4, 499--508: the setting on p. 499, the Theorem on p. 500, its proof in section 2 (pp. 501--505), the computations of section 3 (pp. 506--507) and the Addendum on p. 507. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the Theorem, the conclusion $R_n<5/6$ and the Addendum were read clause by clause on the printed pages. The proof in section 2 and the asymptotics of section 3 were not checked. Nothing here is independently reviewed.

Proof pointer

Section 2 (pp. 501--505), with $T=[n/2]$, prescribes the power sums directly: $S_l=1-\alpha$ for $1\le l\le T$ (2) and $S_l=(1-\alpha)+w_l$ for $T+1\le l\le n$ (3), and defines $b_0=1,b_1,\dots,b_n$ from them by the recursion (4). Lemma 1 (pp. 502--503) gives a condition under which $S_{T+1},\dots,S_n$ can be chosen with $|S_l|\le q$ and $b_n=0$; Lemma 2 (p. 503) bounds the terms in that condition; Lemma 3 (p. 504) gives explicit conditions on $\alpha$ under which it holds with $q=1/2$. With $b_n=0$, the roots $z_2,\dots,z_n$ of $Z^{n-1}+b_1Z^{n-2}+\cdots+b_{n-1}$ together with $z_1=1$ have $S_1,\dots,S_n$ as their first $n$ power sums (p. 505). Fixing $|\alpha|/\operatorname{Re}\alpha=5$ and then $|\alpha|$ small enough, the conditions of Lemma 3 hold for all large $n$, and since $|1-\alpha|<1$ and $1/2<1$ the Theorem follows.

Dependencies

Lemmas 1--3 and the Corollary of Lemma 1 of the same paper. The relations between power sums and coefficients displayed on p. 500, which (4) turns into a definition of $b_1,\dots,b_n$ (p. 501), are said there to follow from the Newton--Girard formulas for the polynomial $(Z-z_2)\cdots(Z-z_n)$.

Bears on

• Problem 519: the problem asks whether an absolute $c>0$ exists with $\max_{1\le k\le n}|\sum_i z_i^k|>c$ for all $z_1,\dots,z_n$ with $z_1=1$. Since $R_n$ is the least value of that maximum (under the normalization the paper calls equivalent, p. 499), a constant that works for all large $n$ is below $R_n$ for those $n$; the paper's $R_n<5/6$ for large $n$ therefore excludes every $c\ge5/6$ (an observation of this page). The paper proves no lower bound and does not answer the question; Harcos's $0.69368$ is a reported computation.