Problem 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 $n$,\[\sum_{k\leq n}M_k > n^{1+c}?\]
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 $n$,\[\sum_{k\leq n}M_k > n^{1+c}?\]
Categories:
Analysis Polynomials
Progress
This is Problem 4.1 in [Ha74] where it is attributed to Erdős.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\[\max_{n\leq N} M_n > N^c.\]Erdős (e.g. see [Ha74]) gave a construction of a sequence with $M_n\leq n+1$ for all $n$. Linden [Li77] improved this to give a sequence with $M_n\ll n^{1-c}$ for some $c>0$.
The third question seems to remain open.
Source: erdosproblems.com/119 | Last verified: January 13, 2026