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).



On Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory."
It seems there is a typo in the application of the hyperbola method. Since $b\_n=\sum\_{d\mid n}a\_d$, we have
\begin{align}
\sum\_{n\le x}b\_n &=\sum\_{n\le x}\sum\_{de=n}a\_d=\sum\_{de\le x}a\_d\\
 &=\sum\_{\substack{de\le x\\ d\le y}}a\_d+\sum\_{\substack{de\le x\\ e\le x/y}}a\_d-\sum\_{de\le x\\ d\le y, e\le x/y}a\_d.
\end{align}
If $A(x)=\sum\_{n\le x}a\_n$, then this can be written as
\begin{align}
\sum\_{d\le y}a\_d\left[\frac{x}{d}\right]+\sum\_{e\le x/y}A\left(\frac{x}{e}\right)-A(y)\left[\frac{x}{y}\right].
\end{align}
The author's assumption that $A(x)\ll x^{\delta}$ then gives the last two terms as $O(xy^{\delta-1})$, and the proof can be continued in the usual manner outlined in the text. Although, it appears that a sign was missed later on as well, which has some bearing on the final stated error (see Conrad's answer below).


