Let $\mu$ denote the Möbius function, and let's define the functions $\mu\_{-}$ and $\mu\_{+}$ as follows: $\mu\_{-}(n):=\frac{\mu(n)^{2}-\mu(n)}{2}$ and $\mu\_{+}(n):=\frac{\mu(n)^{2}+\mu(n)}{2}$. Let $M\_{-}$ be the summatory function of $\mu\_{-}$ and $M\_{+}$ the summatory function of $\mu\_{+}$. One has $M(x)=M\_{+}(x)-M\_{-}(x)$, where $M$ is the summatory fonction of $\mu$, and $\hat{M}(x):=M\_{+}(x)+M\_{-}(x)$ is the number of squarefree integers below $x$. One has $\hat{M}(x)=\frac{1}{\zeta(2)}x+O(\sqrt{x})$. Let $\alpha$ and $\beta$ be such that $M\_{-}(x)=\frac{1}{2\zeta(2)}x+O(x^{\alpha})$ and $M\_{+}(x)=\frac{1}{2\zeta(2)}x+O(x^{\beta})$. If one can have $\alpha=\beta$ then these two quantities can be taken equal to $\frac{1}{2}$, and thus one would have $M(x)\ll\sqrt{x}$ which would imply RH.
What are the best known (unconditional) error terms for $M\_{+}$ and $M\_{-}$? Is there any piece of evidence aside RH that they should be the same up to a multiplicative bounded quantity?  

Thanks in advance.



Best error terms for functions related to square free numbers
As I say in the comment, the asymptotics for $M\_{+}$ and $M\_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M\_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M\_{-}(x) = \frac{1}{2 \zeta(2)} x - \frac{1}{2} M(x) + O(x^{1/2})$. These error terms will have the same order of magnitude unless $M(x)$ is small.


If $M(x) \ll \sqrt{x}$ for all $x$, then Gonek's conjecture mentioned in the [paper of Ng](https://arxiv.org/abs/math/0310381) is false. However,
there will probably be infinitely many $x$ for which $M(x)$ is small, and for
these $x$ it will be less clear what the size of the error terms of $M\_{+}$ and $M\_{-}$ will be. (Corollary 1 of Ng's paper, which is very conditional, implies among other things that the logarithmic density of $x$ for which $M(x) \leq \epsilon \sqrt{x}$ tends to zero as $\epsilon \to 0$.)


Finally, the best unconditional bound on $M(x)$ was given by Ivic in his book "The Riemann Zeta-Function" and states that
$$
M(x) = O\left(x \exp\left(-c\_{1} (\log^{3/5} x) (\log \log x)^{-1/5}\right)\right).
$$
(The best conditional bound was given by Soundararajan in 2009.)


