Just trying to figure out what would be the asymptotic relation for the expression
$\sum_{n\leq X}\mu(n)\tau(n)$,
where $\tau$ corresponds to the number of divisors function (often named $\sigma_0$ or just $d$).
I would like to apply it to get an asymptotic formula for the following:
$\displaystyle S_q(X)=\sum_{n\leq X, (n,q)=1}\dfrac{\mu(n)\tau(n)}{n}$?
The coprimality condition adds of course an extra difficulty, but at least I'd like to have an asymptotic for $q=1$, say.
Thanks in advance!