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

$\lim_n \sum_{2 \le x\le n} (-1)^ { \log \log x }$

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

To start, we note that we can reduce

$\sum_{2 \le x\le n} e^{i \cdot \pi\cdot \log \log x } = \sum_{2 \le x\le n} \log(x)^{i \cdot \pi} $. 

Now this looks like it's related to a "more general zeta function" $\zeta (\log (n), -i\pi)$ (As opposed to the "standard" $\zeta(n,-i\pi)$. 

From here, it would seem that the problem is likely out of reach from basically a college undergrad math student. Any textbooks/links for approaching this (hopefully that can ease me into it) would be nice, thanks.


  [1]: https://math.stackexchange.com/questions/315775/on-the-number-of-integers-with-an-even-number-of-distinct-prime-factors/3303295#3303295