Why was the zeta function introduced? 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?
 A: It is all about Prime Numbers. Among few things we know about prime numbers (i.e. prime numbers are infinite), there are many characteristics we do not know about primes. Some of these unproved conjectures and unknown formulas are: 
- Every even number greater $\gt 4$ can be expressed as a sum of two primes. 
- Twin primes {$p\,, p+2 \in \text{prime}$} are infinite. 
- How to determine if a given number is a prime? 
- How many prime numbers exist below a given number? 
 
Between the 16-17th century, there was a famous problem in mathematics called Basel Problem. Basel problem asked to find the precise summation (closed form) of the reciprocals of the squares of all natural numbers:
$$\lim_{n\rightarrow\infty}\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\text{...}+\frac{1}{n^2}\right) = \space\text{?}$$
This problem lasted for about a hundred (100) years until, it had been solved by Euler in 1734. He was the first to find the limit equal to $\pi^2/6$. After solving this famous problem, Euler continues to investigate these kinds of series, and found that:
$$\small\frac{1}{1^x}+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+\text{... etc} = \frac{2^x}{2^x-1}\cdot\frac{3^x}{3^x-1}\cdot\frac{5^x}{5^x-1}\cdot\frac{7^x}{7^x-1}\cdot\frac{11^x}{11^x-1}\cdot\text{... etc}\quad\colon x\gt1$$
This beautiful identity, later called Euler Product, reflects a direct relationship between all integers and all primes. By rephrasing and extending to complex plane, the Zeta function of a complex argument $s$ is defined by:

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s} = \prod_{p\space prime}\frac{1}{1-1/p^s}\quad\colon Re\{s\}\gt1$$
  Where the behavior of Zeta Function is governed by the behavior of Prime Numbers.

 
In other words, the Zeta function gives a way to study the discrete prime numbers using a continues analytic function. In fact, Euler was fascinating by this conversion idea and likewise he introduce the Gamma function:
$$\Gamma(s)=\int_{0}^{\infty} \frac{x^{s-1}}{e^x}\,dx\quad\colon Re\{s\}\gt0$$
as a way of extending the factorial function from discrete integer points to continuous real curve.
$$n!=n\,(n-1)! \quad\equiv\quad \Gamma(x)=(x-1)\,\Gamma(x-1)$$

- John Derbyshire: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem.
- Julian Havil: Gamma: Exploring Euler's Constant.
- Leonhard Euler: Introduction to Analysis of The Infinite.
