Let $$F(x) = \sum_{d\leq x} \frac{\mu(d)}{d} \log \frac{x}{d}.$$

s it possible/feasible to give an elementary proof of the fact that $F(x)= 1 + o(1)$ (and, ideally, $1+O(1/\log x)$, or better)? By "elementary" I mean "using the properties of $\zeta(s)$ only for $\Re(s)\geq 1$, and of preference only for $s$ real". (Call work with $s$ complex, $\Re(s)\geq 1$, "semi-elementary" if you wish.) I'd also need for it to be possible to make the bounds nicely explicit.

[Note: I am well aware of Ramaré's and Balazard's work, which relies on estimates on $\sum_{m\leq x} \mu(m)$ (derived in turn from estimates on $\sum_{m\leq x} \Lambda(m)$). I am looking for (semi-)elementary estimates, in part because I would like something that can be easily adapted to analogous sums.]