Put $L_\tau=y^{-1/2}(\mathbb Z\tau+\mathbb Z)$, a covolume-one lattice, and define
$$
\Theta_{k,\tau}(t)=
\sum_{0\ne z\in L_\tau}\overline z^{,k}e^{-\pi t|z|^2}.
$$
Termwise Mellin transformation in the initial half-plane gives
$$
G_k(\tau,s)=
\frac{\pi^{s+k}y^{-k/2}}{\Gamma(s+k)}
\int_0^\infty\Theta_{k,\tau}(t)t^{s+k-1}\,dt.
$$

Split the integral at $t=1$. The integral over $[1,\infty)$ is entire in $s$ because the theta sum decays exponentially. Apply the <Poisson summation formula for a Euclidean lattice> and the Fourier eigenfunction calculation from part b to the interval $(0,1]$, then substitute $t\mapsto1/t$. This rewrites the small-time integral as another exponentially convergent integral over $[1,\infty)$ plus explicit elementary Mellin terms. Those terms are meromorphic, but for positive even $k$ their apparent poles are canceled by the zeros of $1/\Gamma(s+k)$. The displayed formula therefore continues holomorphically to every $s\in\mathbb C$, proving the <analytic continuation of a weight-k real-analytic Eisenstein series>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 4 / c / Solution

For fixed $\tau$, Mellin transformation of a weighted Gaussian theta series expresses $G_k(\tau,s)$ as a gamma factor times a Mellin integral. Splitting at one and applying Poisson summation to the small-time part continues it to all $s\in\mathbb C$; for positive even $k$, the reciprocal gamma factor cancels the apparent poles.


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