= Solution
For bosonic <Matsubara frequencies> $\omega_n=2\pi nT$, set $a=vk/(2\pi T)$. Applying the <residue theorem> to $\pi\coth(\pi z)/(z^2+a^2)$, whose integer poles reproduce the desired summands, gives
$$
\sum_{n\in\mathbb Z}\frac1{n^2+a^2}=\frac\pi a\coth(\pi a).
$$
It follows that
$$
T\sum_n\frac1{\omega_n^2+v^2k^2}
=\frac1{2vk}\coth\left(\frac{vk}{2T}\right).
$$
Using the area $S_{d-1}=2\pi^{d/2}/\Gamma(d/2)$ of the unit sphere, the <thermal phase fluctuation> becomes
$$
\langle\theta^2\rangle=
\frac{S_{d-1}}{2\chi v(2\pi)^d}
\int_0^\Lambda dk\,k^{d-2}
\coth\left(\frac{vk}{2T}\right).
$$
At small $k$, $\coth(vk/(2T))\sim2T/(vk)$, so the <infrared divergence> is governed by $\int_0 dk\,k^{d-3}$. It diverges for $d=1,2$ and is finite for $d=3$. Therefore short-range systems cannot have true finite-temperature breaking of this continuous symmetry in one or two dimensions, in agreement with the <Mermin-Wagner theorem>, whereas it is allowed in three dimensions. In two dimensions a <Berezinskii–Kosterlitz–Thouless transition> may still produce quasi-long-range order.
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