Biro 2000 upper estimate turan pure power sum problem

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theorem: States that the minimum over normalized complex n-tuples of the largest of the first n power sums has limit superior strictly below one, with the explicit consequences that R_n is below five sixths for large n and that Harcos's choice of parameter gives 0.69368.

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A. Biró, An upper estimate in Turán's pure power sum problem, Indag. Math. (N.S.) 11 (2000), no. 4, 499--508; DOI 10.1016/S0019-3577(00)80018-X (Crossref record read (UTC)). Communicated by Prof. R. Tijdeman at the meeting of September 25, 2000; received July 2000; the issue head is dated December 18, 2000.

The copy read for this card is a scan of the ten printed pages 499--508 with a machine text layer (physical PDF p. $n$ is printed p. $498+n$). The text layer garbles most formulas, so the statements below were checked on the page images. The file name under which it was downloaded marked the copy as author-hosted, but the hosting URL was not recorded. Provenance: downloaded in the repository's survey of September 2026; the download URL was not recorded; 381,577 bytes. No notice is printed on pp. 499--500 or 507--508; the publisher's page could not be read on 2026-10-02 (the DOI resolves to a script-only redirect stub at linkinghub.elsevier.com and ScienceDirect answered HTTP 403), and the Crossref record for DOI 10.1016/S0019-3577(00)80018-X, read 2026-10-02, names only the publisher's own terms, Elsevier's text-and-data-mining and open-archive user licenses, and no Creative Commons license, every other right reserved.

Read status: claims checked. The Theorem (p. 500), the conclusion $R_n<5/6$ for large $n$ (p. 507) and the Addendum (p. 507) were read clause by clause on the page images; the proof in section 2 and the computations of section 3 were not read.

Contents

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

$$R_n=\min_{\max_t|z_t|=1}\ \max_{1\le j\le n}|S_j|,$$

where the minimum is attained by compactness, and the normalization may equivalently be taken as $z_1=1$ (both noted on p. 499).

• Theorem (p. 500; proof in section 2, pp. 501--505): $\limsup_{n\to\infty}R_n<1$. The paper presents it as the upper-bound analog of Atkinson's 1961 theorem and says it solves Problem 15 of Turán's book.
• Section 3 (pp. 506--507) sketches the precise computation: if a complex $\alpha$ with $|1-\alpha|<1$ and a fixed number $q$ satisfy condition (22), then $\limsup_{n\to\infty}R_n\le\max(|1-\alpha|,q)$ (23); with $\alpha=(1+i)/5$ this gives $R_n<5/6$ for all large $n$ (p. 507).
• Addendum (p. 507): Harcos's computer work with formulas (22) and (23) finds that $\alpha=0.56754+0.54237i$ gives $\limsup_{n\to\infty}R_n<0.69368$; Harcos also observed that the basic identity (14) follows from the inverse Newton--Girard formulas.
• History recorded on pp. 499--500: Turán's 1942 conjecture $R_n>c$, proved by Atkinson in 1961 with $R_n>1/6$; the author's $R_n>1/2$ (Biró 1994) and $R_n>q$ with an absolute $q>1/2$ (Biró 2000); the trivial $R_n\le1$; the Komlós--Sárközy--Szemerédi bounds $R_n<1-1/(250n)$ for $n>n_0$ and $R_n<1-\tfrac13\log n/n$ for infinitely many $n$; the author's $R_n<1-(1-\varepsilon)\log\log n/\log n$ for large $n$ (his "Notes on a problem of Turán", not held); and the numerical conjecture of Cheer and Goldston that $R_n$ has a limit about $0.7$.
• Method (pp. 500--502): the power sums are prescribed as $S_l=1-\alpha$ for $l\le T=[n/2]$ and chosen for $T<l\le n$ so that the numbers $b_l$ defined from them by the recursion (4) satisfy $b_n=0$ (Lemmas 1--3, pp. 502--505); then $z_1=1$ together with the roots of $Z^{n-1}+b_1Z^{n-2}+\cdots+b_{n-1}$ has $S_1,\dots,S_n$ as its first $n$ power sums (p. 505).

Compiled scope

Only the Theorem, the two numerical conclusions and the historical statements above were read; the proof and the computations of section 3 were not checked, and nothing here is independently reviewed. The distinct lower-bound paper of the same year has its own card.

Bears on. #519: the problem asks whether an absolute $c>0$ lies below the largest modulus of the first $n$ power sums whenever $z_1=1$, and $R_n$ is the least value of that modulus. The paper gives upper estimates only: $\limsup R_n<1$ (the Theorem), $R_n<5/6$ for large $n$, so no $c\ge5/6$ works for all large $n$, and Harcos's reported computation $\limsup R_n<0.69368$. It proves no lower bound and does not answer the question.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.