The Modulus of Polynomials with Zeros at the Roots of Unity

for n = 1, 2, 3, .... Note that the polynomial Pn(z) has all its zeros at the kth roots of unity for k = 1, 2, ... , n. So, as n gets larger, this polynomial has zeros that become denser and denser in the unit circle. Does this denseness of zeros force (1) to be true for all n? After our fruitless attempt to find a quick solution to this elementary and naive looking problem, we started giving it more respect and working on it with all the tools in our toolbox. To our pleasure, our findings revealed interesting interactions and connections among different subjects. It is the purpose of this note to share these with the readers.