Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here.
I have a "proof" of the false fact that the Hecke regularized Eisenstein series $$ E_2(\tau) = \lim_{s\to 0} \sum_{(m,n)\ne (0,0)} \frac{1}{(m+\tau n)^2 |m+\tau n|^s}$$ is holomorphic; could someone help me figure out where I'm being too naive?
If we write $$ E_2(\tau, s) = \sum_{(m,n)\ne (0,0)} \frac{1}{(m+\tau n)^2 |m+\tau n|^s},$$ it converges absolutely and uniformly on compact subsets of the upper half-plane for $\text{Re }s>0$. Thus, for $\text{Re }s>0$, $$ \frac{\partial}{\partial \overline{\tau}}E_2(\tau, s) = \sum_{(m,n)\ne (0,0)} \frac{\partial}{\partial \overline{\tau}}\frac{1}{(m+\tau n)^2 |m+\tau n|^s} $$ and the RHS is still an absolutely convergent sum which approaches $0$ as $s\to 0$ since we pick up a factor of $s$ from the antiholomorphic derivative. Thus, by continuity, $ \frac{\partial}{\partial \overline{\tau}}E_2(\tau, 0)=0$, i.e. it's holomorphic.
