Biro 2000 improved estimate power sum problem turan

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theorem: States that some effectively computable absolute constant q greater than one half satisfies max over j at most n of the modulus of the j-th power sum greater than q whenever z_1 equals one, so that R_n exceeds q for every n.

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A. Biró, An improved estimate in a power sum problem of Turán, Indag. Math. (N.S.) 11 (2000), no. 3, 343--358; DOI 10.1016/S0019-3577(00)80003-8 (Crossref record read (UTC)). Communicated by Prof. R. Tijdeman at the meeting of March 27, 2000; received December 1999; the issue head is dated September 25, 2000.

The copy read for this card is a scan of the sixteen printed pages 343--358 with a machine text layer (physical PDF p. $n$ is printed p. $342+n$). The text layer garbles most formulas, so the statement below was checked on the page image. 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; 620,261 bytes. No notice is printed on pp. 343--344 or 357--358; the publisher's page could not be read on 2026-10-02 (ScienceDirect answered HTTP 403), and the Crossref record for DOI 10.1016/S0019-3577(00)80003-8, read 2026-10-02, names only the publisher's own terms, Elsevier's text-and-data-mining user license (https://www.elsevier.com/tdm/userlicense/1.0/) and open-archive user license (https://www.elsevier.com/open-access/userlicense/1.0/), and no Creative Commons license, every other right reserved.

Read status: claims checked. The Theorem and the remark that no concrete value of $q$ is computed (p. 344) were read clause by clause on the page image; the proof (sections 2--4, pp. 345--357) was not read beyond its section structure.

Contents

With $S_j=z_1^j+\cdots+z_n^j$ and $R_n$ the minimum of $\max_{1\le j\le n}|S_j|$ over $n$-tuples with $\max_t|z_t|=1$ (equivalently $z_1=1$; abstract and p. 343):

• Theorem (p. 344; proof pp. 344--357): one effectively computable absolute constant $q>1/2$ bounds $\max_{1\le j\le n}|S_j|$ strictly from below for every $n$ and all complex $z_1,\dots,z_n$ with $z_1=1$; hence $R_n>q$ for every $n$. The paper does not compute a concrete value of $q$ but says the steps of the proof would allow it (p. 344).
• Context (pp. 343--344): Turán's Problem 12 asks for the best constant $c$ with $R_n>c$; the previous best lower bound was $R_n>1/2$ (Biró 1994, Theorem 1); the trivial $R_n\le1$; the Komlós--Sárközy--Szemerédi upper bound; the author's forthcoming $R_n<1-(1-\varepsilon)\log\log n/\log n$ for large $n$, with any fixed $\varepsilon>0$; and the Cheer--Goldston numerical evidence that $R_n$ decreases to a limit about $0.7$.
• Method (pp. 344--345): the proof assumes $|S_j|\le q$ for $1\le j\le n$ with $1/2<q<q_0<1/\sqrt2$ and derives a contradiction for $q$ close enough to $1/2$; it examines when "asymptotic equality" could hold in the 1994 argument, using the Newton--Girard formulas (3), (4) for the coefficients of $\prod_{t=2}^n(Z-z_t)$ and new formulas (14) obtained by summing (3), which involve $c_t=1+b_1+\cdots+b_t$. Sections: 2 auxiliary lemmas (Lemmas 1--6, with Corollaries 1--2 of Lemma 3), 3 new formulas (14) with Lemmas 7--8, 4 proof of the theorem, ending on p. 357 with the contradiction (47).

Compiled scope

Only the Theorem, the remark on $q$ and the introductory statements were read; the proof was not checked, and nothing here is independently reviewed. The distinct upper-estimate paper of the same year has its own card: Biró 2000 (upper estimate).

Bears on. #519; the Theorem proves the problem's existence statement with an effectively computable absolute constant $q>1/2$, whose value the paper does not compute.

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