Problem 119

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claims/: The 4 claim pages of Problem 119, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $n$,

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

Status. 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 [[problems/polynomials/E0119/claims/2026_08_29_korsky|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.

Source. erdosproblems.com/119, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #119, https://www.erdosproblems.com/119.

References.

• [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: erdos_1997_some_unsolved_problems.
• [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.

Formalization. Statement in formal-conjectures, whose three parts are marked solved there with links to a Lean proof in the lean-proofs repository; that proof and a second Lean development posted on the site's thread are recorded on the claim page of the first Korsky bound at their pinned locations; both are unbuilt and unaudited by this corpus.

Current assessment

The question, as the site states it (page last edited 2026-09-01), asks three things of $M_n=\max_{\lvert z\rvert=1}\lvert\prod_{i\le n}(z-z_i)\rvert$ for a sequence on the unit circle: whether $\limsup M_n=\infty$, whether $M_n>n^c$ infinitely often for some $c>0$, and whether $\sum_{k\le n}M_k>n^{1+c}$ for all large $n$ for some $c>0$, the last carrying the prize offered in [Er97f]. The site records the problem as Problem 4.1 of Hayman's 1974 collection [Ha74], attributed to Erdős. The three questions are nested: the third implies the second, the second the first. All three are answered yes, so the corpus records the outcome as proved, each answer accepted on its claim page. Wagner [Wa80] answered the first with $M_n>(\log n)^c$ infinitely often, accepted on Wagner 1980; Beck [Be91] answered the second with $\max_{n\le N}M_n>N^c$ for all $N$, with absolute constants whose value is not known (the zbMATH review Zbl 0747.11031 states the quantifiers), accepted on Beck 1991; both are refereed and credited by the site. The third was answered in July 2026 by Samuel Korsky working with GPT 5.6-Pro: $\sum_{k\le N}M_k\gg N^{5/4}/\sqrt{\log N}$, by smoothing with a Fejér kernel at the next point and summing, which turns the one-sided maxima into a pair energy that Fourier expansion controls. The site's thread marks that claim accepted as correct and its curator posted the full argument; it is not refereed, and the corpus accepts it on the site's documented acceptance, on Korsky 2026. The same claimant's later bound $(e^{-1/2}+o(1))N^{3/2}$, believed sharp, is accepted on the site's crediting sentence (page last edited 2026-09-01), which states the resolution at the strength $n^{3/2-o(1)}$, that bound's exponent, on Korsky 2026 (second claim); the thread's acceptance mark is on the earlier claim, and the write-up is an unrefereed shared PDF. On the other side, Erdős's construction with $M_n\le n+1$ for every $n$ and Linden's [Li77] with $M_n\ll n^{1-c}$ show that the exponent in the second question is below $1$, and the claimant's numerical construction with $\sum_{k\le N}M_k$ of order about $N^{1.533}$ suggests $3/2$ as the true exponent of the third; the exact order of $\sum_{k\le n}M_k$ and of the exponent in the second question remain open. Two Lean developments of the accepted argument exist, one posted on the thread and one in the lean-proofs repository, the latter linked from the formal-conjectures statements of all three parts. Proof coverage: none; both Lean developments are unbuilt and unaudited by this corpus, and the proofs in [Wa80], [Be91] and the two write-ups are recorded from the site's account and the thread, unchecked.

Search scope: the site's problem page as exported (last edited 2026-09-01), its proof-claims thread with both comment threads (as of 2026-10-07), the community database entry (solved with a Lean marker as of its last update, on 2026-08-23), the formal-conjectures file and the linked lean-proofs file at its pinned commit, an arXiv search for a write-up by the claimant (none found), and the zbMATH review of [Be91] (Zbl 0747.11031) for its quantifiers; no OpenAI release item names this problem. MathSciNet was not searched and X was not used.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1959_product
• erdos_1959_product / conjecture_p30
• erdos_1997_some_unsolved_problems