A higher order Levin-Feinleib theorem

Abstract

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 $O((\log X{)}^{\kappa -h-1+\epsilon })$ (for any positive $\epsilon >0$) as soon as we have ${\sum }_{p\le Q}f(p)(\log p)/p=\kappa \log Q+\eta +O(1/(\log 2Q{)}^h)$ for a non-negative $\kappa$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.