Write $\tau=x+iy$ and $\sigma=\operatorname{Re}s>1$. The positive-definite quadratic form
$$
Q_\tau(m,n)=|m\tau+n|^2
$$
is bounded below by $C_\tau(m^2+n^2)$ for some $C_\tau>0$. Hence
$$
\sum_{(m,n)\ne(0,0)}
\left|\frac{y^s}{|m\tau+n|^{2s}}\right|
\leq y^\sigma C_\tau^{-\sigma}
\sum_{(m,n)\ne(0,0)}(m^2+n^2)^{-\sigma},
$$
and the last two-dimensional <lattice sum> converges for $\sigma>1$. Thus the <nonholomorphic Eisenstein series> converges absolutely.

For $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma(1)$,
$$
\operatorname{Im}(\gamma\tau)=\frac{y}{|c\tau+d|^2}
$$
and
$$
m\gamma\tau+n
=\frac{(ma+nc)\tau+(mb+nd)}{c\tau+d}.
$$
The map $(m,n)\mapsto(ma+nc,mb+nd)$ permutes $\mathbb Z^2\setminus\{0\}$, so absolute convergence permits reindexing and gives $G(\gamma\tau,s)=G(\tau,s)$. Hence $G(\tau,s)\in W_0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 4 / b / Solution

{title2=$G(\tau,s)$}

For $\operatorname{Re}s>1$,
$$
G(\tau,s)=\sum_{(m,n)\ne(0,0)}\frac{\operatorname{Im}(\tau)^s}{|m\tau+n|^{2s}}
$$
is an absolutely convergent modular-invariant function.


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 Incoming links: Nonholomorphic Eisenstein series