Riemann Zeta Function and Analytic Continuation

This is a great video that I came across about the Riemann zeta function and analytic continuation.

WolframAlpha explains analytic continuation as follows:

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is to a complex analytic function determined near a point z_0 by a power series:

Such a power series expansion is in general valid only within its radius of convergence. However, under fortunate circumstances (that are very fortunately also rather common!), the function f will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. This allows, for example, the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic functions from the real line R to the entire complex plane C.