Convergence of the Eisenstein series Consider the complex semiplane $\mathcal H=\{z\in\mathbb C\,:\,\Im(z)>0\}$, and lets indicate with $\tau$ the elements of $\mathcal H$. Serre's book "A course in arithmetic" says that the serie of function
$$G_k(\tau)=\!\!\!\sum_{\substack{(m,n)\in\mathbb Z\times\mathbb Z,\\(m,n)\neq(0,0)}}\!\!\!{(m\tau+n)}^{-k}$$
converges normally on compact sets of $\mathcal H$ if $k\ge 3$. I have proved that:
$$\sum_{\substack{(m,n)\in\mathbb Z\times\mathbb Z,\\(m,n)\neq(0,0)}}\!\!\left|{(m\tau+n)}^{-k}\right|\le \frac{8}{r^k}\sum_{n=1}^{\infty}\frac{1}{n^{k-1}}\,\,<\infty$$
for all $\tau\in\mathcal H$ with $r\in\mathbb R$. So $G_k(\tau)$ is (pointwise) absolutely convergent in $\mathcal H$, but i don't understand why this serie is normally convergent on compact sets.
 A: Let $\rho = \frac{1 + \sqrt{-3}}{2}$.
$\rho\bar\rho = 1$.
$\rho + \bar\rho = 1$.
Hence $|m\rho - n|^2 = (m\rho - n)(m\bar\rho - n) = m^2 -mn + n^2$ for $(m, n) \in \mathbb{Z}^2$.
Let $D = \{z \in \mathcal{H}; |z| \ge 1, |Re(z)| \le 1/2\}$.
Let $z \in D$.
Let $(m, n) \in \mathbb{Z}^2$.
$|mz + n|^2 = (mz + n)(m\bar z + n) = m^2z\bar z + 2mnRe(z) + n^2 \ge m^2 -mn + n^2 = |m\rho - n|^2$
Hence $|mz + n|^{2k} \ge |m\rho - n|^{2k}$ for $k \ge 1$.
On the other hand, $\sum_{(m,n)\in \mathbb{Z}^2 - (0,0)} 1/|m\rho - n|^s$ converges for $s > 2$
Hence $G_{2k}(z) = \sum_{(m,n)\in \mathbb{Z}^2 - (0,0)} 1/|mz + n|^{2k}$ converges normally on $D$ for $k > 1$.
We fix $k > 1$
Let $\sigma \in SL_2(\mathbb{Z})$.
Let $K$ be a compact subset of $\sigma(D)$.
Let $z \in K$.
Suppose $\sigma^{-1} = \left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right)$.
$m\sigma^{-1}(z) + n = m(az + b)/(cz + d) + n = ((ma + nc)z + mb + nd)/(cz + d)$ for $(m,n) \in \mathbb{Z}^2$.
Hence $G_{2k}(\sigma^{-1}(z)) = (cz + d)^{2k} G_{2k}(z)$.
Hence $G_{2k}(z) = (cz + d)^{-2k} G_{2k}(\sigma^{-1}(z))$.
$Im(\sigma^{-1}(z)) = Im(z)/|cz + d|^2$
Since $K$ is compact, there exists $M > 0$ such that $|Im(\sigma^{-1}(z))| \le M$ for every $z \in K$.
Since $K$ is compact, there exists $\delta > 0$ such that $|Im(z)| \ge \delta$ for every $z \in K$.
Hence $1/|cz + d|^2 \le \frac{M}{\delta}$ for every $z \in K$.
Hence $|G_{2k}(z)| \le (\frac{M}{\delta})^k |G_{2k}(\sigma^{-1}(z))|$ for every $z \in K$.
Hence $G_{2k}(z)$ converges normally on $K$.
Let $L$ be a compact subset of  $\mathcal{H}$.
By theorem 1 of chapter VII of Serre's book "A course in arithmetic",
$L$ can be covered by a finite number of sets of the form $\sigma(D)$, where $\sigma \in SL_2(\mathbb{Z})$.
Since $L \cap \sigma(D)$ is compact, $G_{2k}(z)$ converges normally on $L \cap \sigma(D)$.
Hence $G_{2k}(z)$ converges normally on $L$.
