= Solution
Replace each <XY model> bond weight by its periodic Gaussian approximation. With angles integrated using $d\theta/(2\pi)$, the <Villain model> is
$$
\boxed{Z_V=(2\pi\beta)^{-N_b/2}\int\prod_x\frac{d\theta_x}{2\pi}
\sum_{\{n_{x,\mu}\in\mathbb Z\}}
\exp\left[-\frac1{2\beta}\sum_{x,\mu}n_{x,\mu}^2
+i\sum_{x,\mu}n_{x,\mu}(\theta_{x+\hat\mu}-\theta_x)\right],}
$$
where $N_b$ is the number of bonds. Changing the angle-measure normalization multiplies $Z_V$ by an explicit constant only. The <Poisson summation formula> applied to one bond gives the equivalent winding representation
$$
\frac1{\sqrt{2\pi\beta}}\sum_{n\in\mathbb Z}e^{-n^2/(2\beta)+in\psi}
=\sum_{p\in\mathbb Z}e^{-\beta(\psi-2\pi p)^2/2}.
$$
Thus the same <Villain model> can be written $Z_V=\int\prod_xd\theta_x/(2\pi)\sum_{\{p_e\}}\exp[-\beta\sum_e(\Delta_e\theta-2\pi p_e)^2/2]$. The coefficient $1/\sqrt{2\pi\beta}$ is the PDF's value; the converted TeX incorrectly places $\beta$ outside the square root. The cosine-to-Gaussian replacement is a large-$\beta$ approximation, but the rest of the calculation treats $Z_V$ as the exact model for every positive $\beta$.
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