New proof of the Gaussian integral using the residue theorem with links to the Riemann Zeta function

In this paper the Gaussian integral is proven using contour integration on $\frac{1}{e^{x^2}+1}$ and linking it using a limit to said Gaussian integral. The limit is alsorelated to the Riemann Zeta function using a few manipulations. This new and original proof comes as an addition to the already many pre-existing proofs of the Gaussian integral.