Eisenstein series

Remarks

For $k \ge 3$ the series converges absolutely and locally uniformly on $\mathbb{H}$. So $G_k$ is holomorphic, and it is a modular form of weight $k$ for $\mathrm{SL}_2(\Z)$. With $q = e^{2 \pi i \tau}$, $\sigma_{k-1}(n) = \sum_{d \mid n} d^{k-1}$ (a divisor function) and $B_k$ the $k$-th Bernoulli number: $$E_k(\tau) = 1 - \frac{2k}{B_k} \sum_{n \ge 1} \sigma_{k-1}(n) q^n.$$ Example: $B_4 = -1/30$ gives $E_4 = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$, and $B_6 = 1/42$ gives $E_6 = 1 - 504 \sum_{n \ge 1} \sigma_5(n) q^n$. Non-example: for odd $k$, the terms for $(m, n)$ and $(-m, -n)$ cancel, so the sum is $0$; this is why $k$ is even. For $k = 2$ the double series does not converge absolutely, and the function $E_2$ obtained by summing in a fixed order is not a modular form.