When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for ∑𝑛≤𝑋𝑓(𝑛)/𝑛 with error term O((log𝑋)𝜅−ℎ−1+ε) (for any positive ε>0) as soon as we have ∑𝑝≤𝑄𝑓(𝑝)(log𝑝)/𝑝=𝜅log𝑄+𝜂+O(1/(log2𝑄)ℎ) for a non-negative 𝜅 and some non-negative integer h. The method generalizes the 1967-approach of Levin and Faĭnleĭb and uses a differential equation.