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.)

1$\begingroup$ I know you asked for a reference, not a proof, and I don't know of one. But I will note that $$S_a(N) = \phi(a) \sum_{n\le N} \frac{\phi(an)}{n^2\phi(a)},$$ where the summand is a multiplicative function of $n$. And sums of this type can be evaluated by offtheshelf theorems. See for example equation (A.27) of my paper math.ubc.ca/~gerg/papers/downloads/AFNSVP.pdf , although the literature cited at the beginning of Appendix A might serve you better if you went there directly (particularly if you want an explicit secondary term $C(a)$). $\endgroup$ – Greg Martin Jan 3 '14 at 6:14

$\begingroup$ I can prove it using Perron's formula as well. But I want to save a space in the article. $\endgroup$ – Alexey Ustinov Jan 3 '14 at 7:34
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.