I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, the infinite product doesn't converge to any nonzero number.

It seems that the behavior of $e^{k\alpha \pi i}$ is not quite "predictable". I have no idea how to approach this problem. Thanks in advance!