I am looking for a closed form or an estimation for the sum of a Gaussian sequence expressed as $$ \sum_{x=0}^{N-1} e^{\frac{-a}{N^2} \: x^2} $$
where $a$ is a constant positive integer.
The interesting part is that I have simulated this sequence using MATLAB for $N=0$ to $10^6$ and found that the result is linear with $N$ and that the line slope is inversely proportional to $a$. However, I believe that the term $N^2$ in the denominator is the reason behind this linearity.
Well, $$ \frac{1}{N}\sum_{x=0}^{N-1}\exp\left[-a\left(\frac{x}{N}\right)^2\right]$$ is a Riemann sum for the function $e^{-az^2}$ over the interval $z\in(0,1)$, hence the original sum behaves like $$N\int_{0}^{1}e^{-az^2}\,dz =\frac{N}{2}\sqrt{\frac{\pi}{a}}\,\text{Erf}(\sqrt{a})$$ for any $a>0$ and large enough $N$.