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Claim. There is an absolute $c>0$ such that for every sequence $z_1,z_2,\dots$ on the unit circle,

$$\max_{n\le N}M_n>N^c\qquad\text{for all large }N,$$

where $M_n=\max_{\lvert z\rvert=1}\lvert\prod_{i\le n}(z-z_i)\rvert$. Hence $M_n>n^c$ for infinitely many $n$: if $M_n\le n^c$ for every $n>n_0$, then $\max_{n\le N}M_n\le\max(C,N^c)$ with $C=\max_{n\le n_0}M_n$, which equals $N^c$ for large $N$, against Beck's strict inequality. This is Beck, The modulus of polynomials with zeros on the unit circle: a problem of Erdős, Ann. of Math. (2) 134 (1991), no. 3, 609–651, cited as [Be91] on the problem page; the quantifiers, absolute constants $c,c_0>0$ with $\max_{n\le N}M_n>c_0N^c$ for all $N$ and all sequences, are as the zbMATH review (Zbl 0747.11031) states them, and shrinking $c$ absorbs $c_0$ for large $N$. It answers the second question of Problem 119 yes. The exponent cannot be $1$: Erdős gave a sequence with $M_n\le n+1$ for every $n$ (recorded in Hayman's collection [Ha74]), and Linden [Li77] a sequence with $M_n\ll n^{1-c}$ for some $c>0$. The value of Beck's constant is not known: the site's discussion (the curator's comment of 2026-07-19) records that nobody has worked it out and presumes that Beck did not compute it.

Covers. The second question, that $M_n>n^c$ for infinitely many $n$ for some $c>0$, and with it the first, already answered by Wagner 1980. It does not cover the third question, on the sum $\sum_{k\le n}M_k$, which Korsky 2026 answered.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Annals of Mathematics in November 1991. Reviewed: erdosproblems.com states that the second question was answered by Beck with this bound (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.