Reference request:

Davenport proved that for every fixed $N>1 $ one has $$ \sup_{\alpha \in \mathbb R } \left | \sum_{1\leq n \leq x } \mu(n) \exp(2 \pi i \alpha n )\right | = O_N\left( \frac{x}{(\log x)^N}\right).$$

Does anyone know if the error term has been improved to something like $O(x \exp(-c \sqrt { \log x } ))$, i.e. as the prime number theorem error?