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Claim. For every sequence $z_1,z_2,\dots$ on the unit circle there is a $c>0$ such that

$$M_n=\max_{\lvert z\rvert=1}\Bigl\lvert\prod_{i\le n}(z-z_i)\Bigr\rvert>(\log n)^c$$

for infinitely many $n$; in particular $\limsup M_n=\infty$. This is Wagner, On a problem of Erdős in Diophantine approximation, Bull. London Math. Soc. 12 (1980), no. 2, 81–88, cited as [Wa80] on the problem page. It answers the first question of [[problems/polynomials/E0119/_index|Problem 119]], which the site records as Problem 4.1 of Hayman's 1974 collection [Ha74], attributed there to Erdős.

Covers. The first question only, that $\limsup M_n=\infty$. It does not reach a power of $n$, which is the second question, answered by Beck 1991, nor the sum of the third question, answered by Korsky 2026.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Bulletin of the London Mathematical Society in March 1980. Reviewed: erdosproblems.com states that the weaker conjecture $\limsup M_n=\infty$ was proved by Wagner with the bound $(\log n)^c$ infinitely often (page last edited 2026-09-01), which the corpus counts as documented independent acceptance by the site's curator, T. F. Bloom (erdosproblems.com). Proof coverage: none; the paper is not held in the library.