When restricted to some non-negative multiplicative function, say $f$, boundedon primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n\leq X} f (n)/n$ with error term $\mathcal{O}(({\rm log} X)^{\kappa - h - 1 + \epsilon}$ (for any positive $\epsilon > 0$) as soon as we have $\sum_{p\leq Q} f (p)(\log p)/p = \kappa \log Q +\eta +O(1/(\log 2Q)^h)$ for a non-negative $\kappa$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fa\u{i}nle\u{i}b and uses a differential equation.
