>>10999341So, originally the zeta function was defined by the formula for positive real values of s. This is the so called Euler Zeta Function, and for all values of s that make the series converge it defines a function of s.
Given a function f(z), an analytical extension of g(z) of f is another function such that f(z) = g(z) everywhere where the equality makes sense but g is defined for more values than f. So if is given for a subset of the reals then the Riemann Zeta Function is it's analytical continuation.
So for all values of s where or series converges we have that , but is defined for negative numbers and complex values too.
Also, . Some people denote by the original sum (what is an abuse of notation), and when you plug s = -1 in the sum it becomes the sum of the naturals. But what they meant to say is that , not that .
It's just that math folks like to cut corners when writing
