Problem 119

Erdős Problem #119: declared status 'open'. Formalized: yes. 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 $n$,$$\sum_{k\leq n}M_k > n^{1+c}?$$ Current best: The second question was answered by Beck [Be91], who proved that there exists some $c>0$ such that$$\max_{n\leq N} M_n > N^c.$$Erd\H{o}s (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 $c>0$. Prize: $100. Tags: analysis, polynomials.