I'm trying to prove that the series
$$
\sum\_{\omega\in \mathbb{Z}(i)^\*} |\omega|^{-2}
$$ 

The problem can be viewed as the sum above the fundamental region $\Omega^\* = \{m\omega\_1 + n\omega\_2 : m,n\in \mathbb{z}\}\backslash\{0\}$ for $\omega\_1 = 1$ and $\omega\_2 = i$. Note: $\mathbb{Z}(i) = \{m + ni : m,n\in\mathbb{Z}\}$ and $\mathbb{Z}(i)^\*=\mathbb{Z}(i)\backslash\{0\}$.

Any hint? I thought it will suffice to use that for $m\geq n > 0$ we have $|m+ni|\leq 2m$, but I get nothing from that... Thank you.