Are there any tight bounds, or any nontrivial ones actually, known for, with the lack of a better name, the Alternating Mertens Function: $$ S(n) = \sum_{k=1}^{n} \left((-1)^k \mu\left(k \right)\right) $$
Edit:
Thanks to i707107. As stated in the answer here, A convergence problem： splitting a double sum , this sum has a more general asymptotic as: $$ S_\theta(n) = \sum_{k=1}^{n} \left(e^\left(2 \pi i \theta k \right) \mu\left(k \right)\right) $$
For which, $ S(n) $ is just a special case where $ \theta = \frac{1}{2} $. It turns out due to a result by H. Davenport that $ S_\theta(n) $ is $ O(x/\ln^\mathcal{E}(x)) $ for all real $ \theta$ and all $ \mathcal{E} > 0 $.
My question is for a more improved bound though, as this bound is not too much greater thank that of $ O(x) $.