Let
$$S\_a(N)=\sum\_{n\le N}\frac{\varphi(an)}{n^2}.$$
The usual machinery gives an asymptotic formula
$$S\_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi\_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\varepsilon}),$$
where $C(a)$ some complicated function and
$$\varphi\_+(a)=a\prod\_{p\mid a}\left(1+\frac1p\right).$$
Is it possible to give a reference on this asymptotic formula? (My proof is rather long.)



On a sum involving Euler totient function
I have found a proof of more general formula in the book *Postnikov, A. G. Introduction to analytic number theory American Mathematical Society, 1988,* (section 4.2). This proof is simple but it has a small mistake inside. (For arithmetic progression starting from $0$ this mistake vanishes.)


Please give more references if you know ones.


