I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum\_{n\in\mathbb Z}f(x+n)$), the resulting function is *remarkably* constant: that is, the function differs from its average value by less than one part in $10^8$. The significance of translating by multiples of 1 is that the inflection point of the Gaussian occurs at $\pm1$. 


I attempted to repeat this for $g(x)=e^{-3x^4/4}$ which also has an inflection point at $\pm 1$, and found that the resulting sum is constant to one part in 200. 



> 
> Can anyone offer any conceptual explanation for the remarkable degree of constancy of $F(x)$, or is this just a fluke?
> 



Sum of Gaussian pdfs
First of all this has nothing to do with the inflection point of $e^{-\alpha x^2}$. According to Poisson summation formula (see Whittaker, Watson, Modern analysis, chapter 21.51)
$$
\sum\_{n=-\infty}^\infty e^{-\alpha (x-n)^2}=2{\sqrt{\frac{\pi}{\alpha}}}\left(1+2\sum\_{n=1}^\infty e^{-\frac{\pi^2}{\alpha}n^2}\cos2\pi n x\right),\quad \text{Re}~\alpha>0.\tag{1}
$$
From this formula one can see that when $\alpha>0$ is small, the Fourier coefficients $a\_n=2{\sqrt{\frac{\pi}{\alpha}}}e^{-\frac{\pi^2}{\alpha}n^2}$ of the resulting function decreases rapidly with increasing $n$. When $\alpha=1/2$ one has
$$
\frac{a\_1}{a\_0}=2e^{-2\pi^2}\approx 5.4\times 10^{-9}.
$$


I don't know any conceptual explanation for this in mathematics, but there is such an explanation that comes from physics. Consider a quantum particle with mass $m$ on a ring of radius $a$. We assume the ring is pierced with magnetic flux $\phi$. We want to calculate partition function of this system at temperature $T>0$ in two different ways. 


On the one hand, it is known that the energy spectrum of the particle is given by
$$
E\_n=\frac{\hbar^2}{2m^2}\left(n+\frac{\phi}{\phi\_0}\right)^2,
$$
where $\phi\_0$ is the so called [magnetic flux quantum](https://en.wikipedia.org/wiki/Magnetic_flux_quantum). Then partition function is given as the [Gibbs sum](https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics))
$$
Z=\sum\_{n=-\infty}^\infty e^{-E\_n/T}=\sum\_{n=-\infty}^\infty e^{-\frac{\hbar^2}{2m^2T}\left(n+{\phi}/{\phi\_0}\right)^2}.\tag{2}
$$
Here one immediately recognizes the sum analogous to the LHS of $(1)$. 


On the other hand, partition function is related to the trace of the [density matrix](https://en.wikipedia.org/wiki/Density_matrix) $\rho\_{\theta\_1,\theta\_2}$, i.e. to $\int\rho\_{\theta,\theta}d\theta$. It is possible to calculate the density matrix of this system in imaginary time representation by solving a certain differential equation. Details of this calculation can be found for example in this [book](https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwishpPqjY7XAhWsCpoKHdHqAAMQFggmMAA&url=http%3A%2F%2Fserver.bling-my-thing.com%2Fchapter_4_aharonov_bohm_effect_and_geometric_phase.pdf&usg=AOvVaw3NVQUydLRnsX6rZAW6avM3), chapter 4.3. Consider first the more simple case of unbounded line $-\infty<\theta<+\infty$; then the answer is
$$
\rho\_{\theta\_1,\theta\_2}=Ce^{-\frac{mTa^2(\theta\_1-\theta\_2)^2}{2\hbar^2}},
$$
where $C$ is some normalization constant. Magnetic flux does not enter this expression because there is not any nontrivial loop in the system. On a ring there is a nontrivial loop, and let $n$ be the winding number. It is known that the partition function is a sum over all homotopy classes multiplied by corresponding phase factor. In this case the phase factor comes from the magnetic flux ([Aharonov-Bohm phase](https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect)) and equals $e^{2\pi in\phi/\phi\_0}$.
As a result we have
\begin{align}
Z&=\sum\_{n=-\infty}^\infty e^{2\pi in\phi/\phi\_0}\int\rho\_{\theta,\theta+2\pi n}d\theta\\
&=2\pi C \sum\_{n=-\infty}^\infty e^{-\frac{2\pi^2mTa^2}{\hbar^2}n^2+2\pi in\phi/\phi\_0}.\tag{3}
\end{align}
Combining $(2)$ and $(3)$ one obtains the transformation in $(1)$. 


One can see from this physical interpretation that when the temperature $T$ increases, then the higher Fourier harmonics decrease. This corresponds to the physical intuition that the higher the temperature the more chaotic the system becomes and the effect of the Aharonov-Bohm phase averages out due to thermal fluctuations. 


