I'm interested in how one would estimate the growth rate of
$$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$
I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head (like using the fact that the density of the sqrfree integers is $6/\pi^2$) but perhaps there's a smarter way. If there was a way of truncating
$$h(s)=\sum_{k=1}^{\infty}\frac{\mu^2(k)\log(k)}{k^s}$$ We could use that, and I think I have seen something of this nature before, that is, truncating Dirichlet series. But anyway, what is your approach?