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

Write $\tau =x+iy$ and $\sigma =\mathrm{Re}s>1$. The positive-definite quadratic formis bounded below by $C_{\tau }(m^2+n^2)$ for some $C_{\tau }>0$. Henceand the last two-dimensional lattice sum converges for $\sigma >1$. Thus the nonholomorphic Eisenstein series converges absolutely.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,andThe 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 $G(\tau ,s)\in W_0$.