# Error term in Davenport's sum $\sum_{n \leq x } \mu(n) \exp(2 \pi i \alpha n )$

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?

• 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. – AndreyF Dec 11 '20 at 17:57

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.