The Riemann zeta function and analytic continuation, how many solutions are possible to his analytic continuation?

Question

When Riemann developed the zeta function in the complex plane, he used a process called analytic continuation. He found only one solution. Does this mean that there was only one analytic solution to find? I have been told that if there are any solutions to the analytic continuation, there would be only one unique solution, but this seems to be opinion. I cannot find a proof of this suggestion, but would like find/ develop such a proof as I am teaching number theory for a class of adults at the University of the Third Age, Cambridge, UK.

Answer 1

Yes, analytic continuation works that way. In fact, if $f(z)$ is an analytic function defined on $\mathbb C \setminus \{1\}$ that agrees with $\zeta(z)$ on a set with a limit point, then $f(z) = \zeta(z)$, the extension found by Riemann. Certainly the set of real numbers ${}\gt 1$ is a set with a limit point.

See the Identity Theorem in any textbook on complex variables.

It is easy to show there is at most one extension. But the hard part (done by Riemann) shows that there is an extension.

The example of extending $\log z$ to $\mathbb C \setminus \{0\}$ is a case where there is no extension at all. But still, there is at most one extension, which is easy to prove.