A question regarding change of index of summation in analytic number theory

Question

I am trying exercises of Apostol's Dirichlet Series and Modular Functions in Number Theory and I am unable to get past an argument in this question.

!Original Question statement of book]1

->!Defination of $G_2(\tau)$ ]2

!Equation (12)]3

!What I tried in the question/My attempt]4

Can someone please help how to complete it!!

Answer 1

I looked through the notes and you have a double sum in two variables $n,r$ in the next to last line as you sum on $n$ in a sum on $r$, so the sum is: $S=\sum_{n \ge 1}\sum_{r=1}^{\infty} r e^{2πirn\tau} $

Note that in the original sum with $\frac{1}{(n\tau+m)^2}$ terms you can extend the sum to all nonzero integers in $n$ using the trick that $\frac{1}{(n\tau+m)^2}=\frac{1}{(-n\tau-m)^2}$ and the sum in $m$ is on full integers, so you get double and take care of the coefficient $2$ in equation $(49)$ but in the sum $S$ above $n$ must be positive for convergence - so this may be a source of confusion here, as $n$ is just a summation variable in both with similar but not quite identical meanings)

But now you switch variables to $q=nr$ (allowable by compact absolute convergence when $\Im \tau >0$) remembering that in $S$ above you can have only $n \ge 1$ as the $n$ negative appears only in the original sum you start with where the terms are $\frac{1}{(n\tau+m)^2}$), it is clear that $\sum_{r|q} re^{2πiq\tau}=\sigma(q)e^{2πiq\tau}$ so you get that $S=\sum_{q \ge 1}(\sum_{(r,n), rn=q}r e^{2πiq\tau})=\sum_{q \ge 1}\sigma(q)e^{2πiq\tau}$ and then changing variables back to $q=n$ you are done!

Note that the $n$ in the end (in the terms $\sigma(n)e^{2πin\tau}$) and the $n$ in the beginning (in the terms $\frac{1}{(n\tau+m)^2}$) have completely different meanings as they are just summation variables so that may be another source of confusion

Summarizing - the computations are straightforward per the hint and compact absolute convergence for $\Im \tau >0$ allows switching double sums and changing variables in them almost at will, but the usage of the summing variable $n$ in several different contexts with different meanings may lead to confusion