In his 'Introduction to modular forms', Don Zagier deals with the Hecke's trick which I don't really understand :
Let $$G_2(\tau)=-\frac{1}{24}+\sum_{n=1}^{+\infty}{\sigma_1(n)q^n}$$ and $$G_2^*(\tau)=\frac{-1}{8\pi^2}\lim_{\epsilon \to 0}{\left(\sum_{m,n}{'}{\frac{1}{(m\tau+n)^2 |m\tau+n|^{\epsilon}}}\right)}$$ It is said that with Poisson formula, one can find that $G_2^*(\tau)=G_2(\tau)+(8\pi v)^{-1}$ where $\tau=u+i v \in \left\{\Im(\tau)>0\right\}$.
Can someone develop the proof or give a reference ?
Thanks a lot !