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, $A(x)=O(e^{-c\log x})$ 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.

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.