Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum\_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does there exist any better bound conditionally or unconditionally?


I am expecting a result like $A(x)=O(\frac1{\sqrt x})$ if Riemann Hypothesis is assumed. Is this true? What I could get is up on truth of RH $A(x)=O(x^{-1/2+\epsilon})$, which is not very hard to prove.


Any reference regarding conditional (RH) bound of $A(x)$ will be highly appreciated. Thanks.



Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$
As Alexey has pointed out the problem can be reduced, via [summation by parts](http://en.wikipedia.org/wiki/Summation_by_parts), to understanding the asymptotic of Mertens' sum 
$$M(x) := \sum\_{n\leq x} \mu(n).$$
Conditional on the Riemann hypothesis, the best bound to date on Mertens' sum, [due to Soundararajan](http://www.ams.org/mathscinet-getitem?mr=2542220) in 2009, is
$$M(x) \ll x^{1/2} e^{(\log(x))^{1/2} (\log\log(x))^{14}}.$$
In particular, this implies
$$\sum\_{n\leq x} \frac{\mu(n)}{n} \ll \frac{e^{(\log(x))^{1/2} (\log\log(x))^{14}}}{x^{1/2}}.$$


