Are there results, or can someone point me to reading, on computations of the following:

$\lim_n \sum_n cis(\pi \cdot \log \log n)$

It would seem that there is reason to believe this should be something like $-\log(n)$. For example, see the graphic on this question which is the visual representation of $\sum_x (-1)^{\omega(x)}$ which for large $x$ is like $\sum_x (-1)^{\log \log (x)} = \sum_x cis(\pi \cdot \log \log (x))$. On expectation, this looks something like $- \log n$.

I dont know where to start, sources would be nice, thank you!