= Iterated Eisenstein summation in weight two
{title2=$G_2(z)=\sum_m\sum_n'(mz+n)^{-2}$}
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 $-4\pi^2\sum_{r\geq1}r q^{mr}$; negative $m$ gives the same contribution. Including the $m=0$ term yields $\pi^2/3-8\pi^2\sum_{n\geq1}\sigma_1(n)q^n$. The full lattice series does not have <absolute convergence>, so this order must not be replaced by arbitrary rearrangement.
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