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OPEN - $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.

Additional thanks to: Winston Heap