Iterated Eisenstein summation in weight two ($G_2(z)={\sum }_m{\sum }_n^{\prime }(mz+n{)}^{-2}$)

In the displayed iterated order, with the $n$ sum evaluated first, $G_2(z)=({\pi }^2/3)E_2(z)$. The cosecant partial-fraction identity ${\sum }_n(w+n{)}^{-2}={\pi }^2{\csc }^2(\pi w)$ gives the positive-$m$ contribution $-4{\pi }^2{\sum }_{r\ge 1}r{q}^{mr}$; negative $m$ gives the same contribution. Including the $m=0$ term yields ${\pi }^2/3-8{\pi }^2{\sum }_{n\ge 1}{\sigma }_1(n)q^n$. The full lattice series does not have absolute convergence, so this order must not be replaced by arbitrary rearrangement.