Let $n\ge 1$ and $f(n)$ be maximal such that for any integers $1\le a_1\le \cdots \le a_n$ we have$$\underset{|z|=1}{\max }\left|\prod _i(1-z^{a_i})\right|\ge f(n).$$Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that$$\log f(n)\gg n^c?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let [ P_{(a_1,\dots,a_n)}(z):=\prod_{j=1}^n(1-z^{a_j}),\qquad |z|=1, ] and [ M(a_1,\dots,a_n):=\max_{|z|=1}|P_{(a_1,\dots,a_n)}(z)|. ] Your “maximal” $f(n)$ is exactly [ f(n)=\inf_{1\le a_1\le\cdots\le a_n} M(a_1,\dots,a_n), ] the classical Erdős–Szekeres “pure power product” extremal function.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.