Eisenstein series as a double series

Example

Let $\tau$ be a point of the upper half-plane and let $k \ge 3$ be an integer. The Eisenstein series $G_k(\tau) = \sum_{(m,n) \ne (0,0)} (m\tau + n)^{-k}$ is a double series indexed by $\Z^2 \setminus \set{(0,0)}$. It converges absolutely. Check: since $1$ and $\tau$ are linearly independent over $\R$, there is $c > 0$ with $\abs{x\tau + y} \ge c \max(\abs{x}, \abs{y})$ for all real $x, y$. For each integer $r \ge 1$ there are exactly $8r$ pairs $(m, n)$ with $\max(\abs{m}, \abs{n}) = r$. So $\sum \abs{m\tau + n}^{-k} \le \sum_{r \ge 1} 8r \, (cr)^{-k} = 8 c^{-k} \sum_{r \ge 1} r^{1-k}$, which is finite because $k - 1 \ge 2$. By absolute convergence, the sum does not depend on the order of the terms. Summing by rows, by columns or over growing squares gives the same value. For odd $k$ the terms for $(m,n)$ and $(-m,-n)$ cancel and $G_k = 0$, so the interesting cases are the even $k \ge 4$.

For $k = 2$ the same count gives the divergent bound $\sum 8r(cr)^{-2}$, and indeed the double series is not absolutely convergent. Summing first over $n$ and then over $m$ still gives a value, but this function is not a modular form. The order of summation matters here.