TITLE: Error term in Davenport's sum $\sum_{n \leq x } \mu(n) \exp(2 \pi i \alpha n ) $
QUESTION [9 upvotes]: 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?

REPLY [13 votes]: If there's a bad Landau--Siegel zero $\mod q$, then the Möbius function behaves a lot like a primitive quadratic character $\mod q$, and then the exponential sum at $\alpha =a/q$ roughly has size $x/\sqrt{q}$ (as would be the case if $\mu(n)$ were replaced by a character).   Siegel's theorem is what enables one to save arbitrary powers of $\log$ in Davenport's result.  If you assume that there are no Siegel zeros, then one can get a refinement as asked.  This was done by Hajela and Smith; and on GRH one can get a power saving.  See the paper of Baker and Harman for details and references.