I assume this question has been considered before, but I can't find an literature on it. Let $\mu(n)$ denote the usual Mobius function and define:

$F(x) : = \sum_{n=1}^{\infty} \frac{\mu(n)}{n}e(nx)$

where $e(x):= e^{2\pi i x}$.

Is $F(x)$ uniformly bounded? And if so, is it continuous?