Holomorphic Eisenstein series of level 1 (reviewed)

For an even integer $k\geq4$, we define the (normalized) holomorphic Eisenstein series of level 1 \[ E_k(z)=\frac{1}{2\zeta(k)}\sum_{(c,d)\ne(0,0)}(cz+d)^{-k}=\sum_{\left(\begin{matrix} a&b\\c&d \end{matrix}\right)\in\ \Gamma_\infty\setminus \SL(2,\Z) }(cz+d)^{-k}, \] where $\Gamma_z=\{\gamma\in\Gamma: \gamma z=z\}$ is the isotropy group of the cusp $z$.

The Eisenstein series $E_k$ are modular forms of weight $k$ and level $1$ on the modular group.

They have the following $q$-expansion: \[ E_k(z)=1-\frac{2k}{B_k}\sum_{n\geq1}\sigma_{k-1}(n)q^n, \] where the $B_k$ are the Bernoulli numbers, $\sigma_{k-1}(n)$ is a divisor function, and $q=e^{2 \pi i z}$.