I have just been told about this result, available as Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory (2nd edition)". It says:
Let $\alpha>0$. Suppose $a_n \ll n^{\alpha}$ and $A(x) \ll x ^{\delta} $ for some $\delta<1$, where $A(x) = \sum_{n\leq x} a_n$. Define $b_n = \sum_{d|n} a_d$. Then one has
$$\sum_{n\leq x} b_n = cx + O\Big(x^{(1-\delta)(1+\alpha)/(2-\delta)}\Big)$$ for some constant $c$.
Does anyone know who first came up with this result, or maybe it's just too straightforward to be attributed to anyone ?
The reason why i'm particularly interested in knowing the originator of this result is that, Murty's proof (which is on pages 262-263 of the aforementioned book) doesn't look quite right to me (but of course, i may be mistaken).