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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?

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    $\begingroup$ Davenport's 1937 result has received new interest due to the fact that it provides a positive example for the Mobius-orthogonality conjecture of Sarnak related to the Chowla conjecture. I am not able to answer your question but have reason to believe that papers by Cellarosi and Sinai (2011 and 2013 maybe) bear directly on your question. If I recall correctly, Cellarosi has a youtube video in which the Davenport result plays a role. I am less certain of the relevance of the paper by Ferenczi, Kulaga-Przymus, and Lemanczyk (~2017) but thought it worth mentioning. $\endgroup$
    – AndreyF
    Commented Dec 11, 2020 at 17:57

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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.

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