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.)

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. 

