OPEN
This is open, and cannot be resolved with a finite computation.
- $100
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 $n$,\[\sum_{k\leq n}M_k > n^{1+c}?\]
The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner
[Wa80], who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.
The second question was answered by Beck
[Be91], who proved that there exists some $c>0$ such that\[\max_{n\leq N} M_n > N^c.\]The third question seems to remain open.
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Additional thanks to: Winston Heap
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #119, https://www.erdosproblems.com/119, accessed 2025-11-15
