A Step-by-step of the Analytic Continuation of the Riemann Zeta Function

Introduction

The Riemann zeta function is widely known (to be the analytic continuation of Euler’s zeta function):

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$$

It is widely known mainly because of the celebrated hypothesis by Riemann that remains unsolved after more than a century’s attempts by mathematicians and 150 million attempts by computers:

Riemann Hypothesis: The non-trivial zeros of $\zeta(s)$...