Problem 119 — Visual explainer (undergrad)
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Problem statement
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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 $n$,\[\sum_{k\leq n}M_k > n^{1+c}?\]

Picture
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[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
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- The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner, who
  show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.
- The second question was answered by Beck, 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 ) gave a construction of a sequence with $M_n\leq n+1$ for all $n$.
- Linden improved this to give a sequence with $M_n\ll n^{1-c}$ for some
  $c>0$.