Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function.

I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the asymptotics for $F$? I can't seem to use any tools that handle sums of the form, $\sum_{n \leq x} f(n)$ where $f:\mathbb{N} \to \mathbb{C}$. Does anyone have any suggestions?