I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form $$ \sum_{n\leq X} a(n) = cX^a(\log X)^m + O\big(X(\log X)^{m-1}\big),$$ i.e. with an error term better than $o(1)$. I have looked into Tenenbaum's book (Theorem 7.13) but both the condition and the expression for the error term seem overly complicated. I also looked into Korevaar's book but couldn't find it there either.
All I managed to get is this article (in french) where Theorem A.1 in Appendix A gives this type of result under some growth condition. The authors say that this is a well-known theorem but they could not find it in the literature, so I wondered where this might appear.