Let $\mu(n)$ be the Mobius function. 
Let us define $\mu^+(n)$ to be $\mu(n)$ if $\mu(n)>0$ and $0$ otherwise. Is there a known asymptotic formula for 
$$
\sum\_{n \leq N} \mu^+(n),
$$ 
and similarly for 
$$
\sum\_{ \substack{n \leq N \\ n \equiv a (\mod q)}} \mu^+(n). 
$$
I would greatly any appreciate references or comments. 
Thank you very much.



Distribution of Mobius function
I'll have a stab at this. The relation 
$$
\sum\_{n\leq X:n\equiv a~(mod~q)} \mu^2(n)=\frac{6}{\pi^2} \prod\_{p|q}
\left(1-\frac{1}{p^2} \right)\frac{X}{q}+E(X,q,a)
$$
where the error term $E$ is $O\_{\varepsilon}\left(\sqrt{X/q}
+q^{\frac{1}{2}+\varepsilon}\right) $ provided $q\leq X^{\frac{2}{3}-\varepsilon},$
together with some recent results (assuming GRH) of the form


$$
\sum\_{n\leq X:n\equiv a~(mod~q)} \mu(n) \ll\_{\varepsilon}
\sqrt{X} \exp\left( (\log X)^{3/5} (\log \log X)^{(16/5)+\varepsilon} \right)
$$


(see the paper on arxiv [here](https://arxiv.org/pdf/1406.7326.pdf)) and the observation $\mu^{+} (n)=\frac{1}{2}(\mu(n)+\mu^2(n))$ suffices to yield an answer.


**Edit** The paper [here](https://arxiv.org/pdf/1603.06758.pdf) gives the bound 
$$
O\left( \frac{X}{q\log^C X}\right)
$$


for any $C>0$, for $X$ large enough.


