This is Problem 4.1 in \cite{Ha74} where it is attributed to Erd\H{o}s.

The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner \cite{Wa80}, who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.

The second question was answered by Beck \cite{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 \cite{Ha74}) gave a construction of a sequence with $M_n\leq n+1$ for all $n$. Linden \cite{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.


References


[Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erd\H{o}s. Annals of Math. (1991), 609-651.

[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.

[Li77] Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math. Soc. (1977), 65--69.

[Wa80] Wagner, Gerold, On a problem of {E}rd\H{o}s in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.