I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function').
I also know the 'Zeta Function' $\zeta(s)$ is defined as:
$$\zeta (s) = \sum_{n=1}^{\infty} {1\over {n^s}}$$
But my question is why and how was this even derived?
I've studied and understood many proofs regarding $\zeta(s)$, such as:
$$\Gamma(s) \zeta(s) = \int_{0}^{\infty} {{u^{s-1}\over {e^u}-1}} \space du$$
$$\zeta(s) = {2^s}{\pi^{s-1}}{sin \bigg({\pi s\over 2}\bigg)}{\Gamma(1-s)}{\zeta(1-s)}$$
But anytime I try search up information regarding the derivation of $\zeta(s)$, all I get is the fact that Leonhard Euler was amongst the first to study it.
Nothing more.
Is there any article I can read that talks about how $\zeta(s)$ came to be?