Problem 256

Erdős Problem #256: declared status 'open'. Formalized: no. Let $n\geq 1$ and $f(n)$ be maximal such that for every $a_1\leq \cdots \leq a_n\in \mathbb{N}$ we have$$\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).$$Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that$$\log f(n) \gg n^c?$$ Current best: Erd\H{o}s proved an upper bound of $\log f(n) \ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Prize: no. Tags: analysis.