information:extraction, structural:scholar_edition, linguistic:paraphrasing
Evaluating Eisenstein series

The same trick can be used to establish the sum of the Eisenstein series: π cot (π z) = lim N → ∞ ∑ n = − N N (z − n) − 1. {\displaystyle \pi \cot(\pi z)=\lim _{N\to \infty }\sum _{n=-N}^{N}(z-n)^{-1}.}

Pick an arbitrary w ∈ C ∖ Z {\displaystyle w\in \mathbb {C} \setminus \mathbb {Z} }. As above, define

g (z):= 1 w − z π cot (π z) {\displaystyle g(z):={\frac {1}{w-z}}\pi \cot(\pi z)}

By the Cauchy residue theorem, for all N {\displaystyle N} large enough such that Γ N {\displaystyle \Gamma _{N}} encircles w {\displaystyle w},

1 2 π i ∮ Γ N g (z) d z = − π cot (π z) + ∑ n = − N N 1 z − n {\displaystyle {\frac {1}{2\pi i}}\oint _{\Gamma _{N}}g(z)dz=-\pi \cot(\pi z)+\sum _{n=-N}^{N}{\frac {1}{z-n}}}

It remains to prove the integral converges to zero. Since π cot (π z) / z {\displaystyle \pi \cot(\pi z)/z} is an even function, and Γ N {\displaystyle \Gamma _{N}} is symmetric about the origin, we have ∮ Γ N π cot (π z) / z d z = 0 {\displaystyle \oint _{\Gamma _{N}}\pi \cot(\pi z)/zdz=0}, and so

∮ Γ N g (z) d z = ∮ Γ N (1 z + 1 w − z) π cot (π z) d z = − w ∮ Γ N 1 z (z − w) π cot (π z) d z = O (1 / N) {\displaystyle \oint _{\Gamma _{N}}g(z)dz=\oint _{\Gamma _{N}}\left({\frac {1}{z}}+{\frac {1}{w-z}}\right)\pi \cot(\pi z)dz=-w\oint _{\Gamma _{N}}{\frac {1}{z(z-w)}}\pi \cot(\pi z)dz=O(1/N)}