Therefore, we have absolute convergence of the original series, and

we know that absolute convergence implies regular convergence and therefore,

the series 1 over n to the s converges.

In fact, one can show that the convergence is uniform so equally fast in

portions of the plane where the real part is strictly bigger than 1.

So, here's 1, if you choose an r that is strictly bigger than 1,

then the convergence is going to be uniform in

the half plane that starts at real part equals r.

One can use that, the fact that uniform convergence and

the fact that the function s goes to 1 over n to the s is

analytic to show that the zeta function is actually

analytic in the entire half plane that starts at 1.

Riemann's seminal contribution to the theory of the zeta function,

and the study of prime numbers is the following theorem.

The zeta function has an analytic continuation into the entire complex

plane with the exception of the number 1.

And this continuation satisfies that zeta of s goes to infinity as s goes to 1.

So far, the zeta function's only defined for a real part of s greater than 1.

So, over here, we have a definition.

Riemann showed that you can extend this

definition to everywhere except the point 1.

The point 1 forms a so-called pole of the zeta function s.

s approaches 1, the zeta function goes off to infinity.

The proof is not easy but a slightly easier construction is to just

construct an extension to the entire right half plane.

So, here this picture, the zeta function s of now is defined for s greater than 1,

and we can actually give the construction that shows you can define it for

s greater than 0 again, except at the point 1.

So, that construction is slightly easier and we'll outline it here.

Again, a motivation in R first.

The sum from n = 1 to some bound uppercase N of 1 over n to

the s again can be viewed as false.

The first term is 1 over 1 to the s.

1 over 1 to the s is this height right here, so

the first term is the area of this first rectangle.