Problem 119

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$ , and for $n\geq 1$ let

p_n(z)=\prod_{i\leq n} (z-z_i).

Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$ .

Is it true that $\limsup M_n=\infty$ ?

Is it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$ ?

Is it true that there exists $c>0$ such that, for all large $n$ ,

\sum_{k\leq n}M_k > n^{1+c}?

Source: erdosproblems.com/119

Analysis
Polynomials

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

The site labels the problem SOLVED (LEAN); the Lean suffix refers to formal proofs by others, recorded on the claim pages, that this corpus has not built or audited. The site credits the first question to Wagner [Wa80], the second to Beck [Be91], and the third, the prize question, to Korsky with GPT 5.6-Pro. The claim pages are Wagner 1980 ( $M_n>(\log n)^c$ infinitely often), Beck 1991 ( $\max_{n\le N}M_n>N^c$ ), Korsky 2026 ( $\sum_{k\le N}M_k\gg N^{5/4}/\sqrt{\log N}$ by a one-page harmonic-analysis argument) and Korsky 2026 (second claim) (the sharper bound $(e^{-1/2}+o(1))N^{3/2}$ , which the site's commentary credits at the strength $n^{3/2-o(1)}$ ); see Current assessment for the evidence.

Be91: Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdős. Annals of Math. (1991), 609-651.
Er97f: Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. Library home: Erdős 1997.
Ha74: Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
Li77: Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math. Soc. (1977), 65-69.
Wa80: Wagner, Gerold, On a problem of Erdős in Diophantine approximation. Bull. London Math. Soc. (1980), 81-88.

Problem 119

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Proof claims

Discussion