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A higher order Levin-Feinleib theorem

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 $\sum_{n \le X}f(n)/n$ with error term $O((\log X)^{κ-h-1+\varepsilon})$ (for any positive $\varepsilon>0$) as soon as we have $\sum_{p\le Q}f(p)(\log p)/p=κ\log Q+η+O(1/(\log2Q)^h)$ for a non-negative $κ$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.