I am curious what are the known bounds on the truncated Dirichlet series $\sum_{N\le n\le 2N}\dfrac{\mu(n)n^{it}}{\sqrt{n}}$.
Naively, we may upper bound this sum by some constant times $\sqrt{N}$. However, can it be proven, whether conditionally on RH or not, that $|\sum_{N\le n\le 2N}\dfrac{\mu(n)n^{it}}{\sqrt{n}}| \ll \dfrac{\sqrt{N}}{t^c}$ for some $c>0$?