---

Claim. For every sequence $z_1,z_2,\dots$ on the unit circle, with $M_k=\max_{\lvert z\rvert=1}\lvert\prod_{j\le k}(z-z_j)\rvert$,

$$\sum_{k\le N}M_k\ge\bigl(e^{-1/2}+o(1)\bigr)N^{3/2},$$

which the claimant believes to be asymptotically sharp. This is the second proof claim registered on the site's thread by Samuel Korsky, on 2026-08-29, with a write-up linked from the thread; the summary credits GPT-5.6 Pro with most of the argument, building on the claimant's own attempt through Hadamard's inequality, and describes the proof as simpler than the earlier one. It answers the third question of Problem 119 with any $c<1/2$, and so gives $M_n>n^{1/2-o(1)}$ for infinitely many $n$, strengthening the accepted answer Korsky 2026, whose exponent is $5/4$. The claimant also reports a greedy construction from $2^{14}$-th roots of unity with $\sum_{k\le N}M_k$ of order about $N^{1.533}$, with the exponent appearing to approach $3/2$ under more computation, and asks for a sequence with $\sum_{k\le N}M_k\le N^{3/2+o(1)}$.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 29 August 2026, giving "GPT-5.6 Pro" as the AI used:

I continued to work on this problem and was able to improve my earlier bound to:

$$\sum_{k \le N}M_k \ge (e^{-1/2} + o(1))N^{3/2}$$

which I believe to be

asymptotically sharp. Amazingly, this proof is even simpler than the last! Unlike my previous $5/4 - o(1)$ exponent bound this improvement was significantly driven by GPT, although an initial (failed) attempt to use Hadamard's Inequality was fully my own, and likely the base on which GPT was able to produce this argument. Notes: I am very interested in a proof of the reverse inequality, namely that there exists some sequence $(z_k)$ for which

$$\sum_{k \le N}M_k \le N^{3/2 + o(1)}.$$

By a greedy-like computation

involving $2^{14}$-th roots of unity I was able to produce such a sequence with exponent $\approx 1.533$, and this exponent seems to shrink towards $1.5$ the more computation is thrown at it.

Depends on. No page of this wiki.

Acceptance. Reviewed: erdosproblems.com labels the problem SOLVED (LEAN) and its commentary (page last edited 2026-09-01) states that the third question was resolved by GPT 5.6 and Korsky, who proved $\sum_{k\le n}M_k\gg n^{3/2-o(1)}$ and hence $M_n>n^{1/2-o(1)}$ for infinitely many $n$; only this claim reaches the exponent $3/2$, the earlier claim giving $5/4$, so the sentence credits this result, and the corpus counts it as documented independent acceptance by the site's curator, T. F. Bloom (erdosproblems.com), who is independent of the claimant. The thread's acceptance mark is on the earlier claim; this one's two comments praise the work and suggest an arXiv preprint without examining the argument. Not refereed: the write-up is an unrefereed shared PDF, and no preprint, journal version or formalization of this bound is recorded. The problem's standing is also fixed by the accepted earlier claim and does not depend on this page.