= Solution
The density is the symmetric sum
$$
g(z)=\frac3{\pi^2\sqrt{2\pi}\sigma}\sum_{n\ne0}\frac{e^{-(z-n)^2/(2\sigma^2)}}{n^2}.
$$
For small $\sigma$, the <normal distributions> are narrow and well separated, giving peaks near each nonzero integer. The largest two peaks lie near $-1$ and $1$; those near $\pm2$ have approximately one quarter of their height, and the envelope falls as $1/n^2$. There is no component centered at zero, and the density there is exponentially small. The left panel shows the resulting smooth comb.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1-smoothed-integer-distribution.png]
{title=Narrow separated peaks and a broad central profile after Gaussian smoothing of heavy discrete tails}
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