Today I did some reading on the Riemann Hypothesis and decided to play around with $\zeta(s)$ a little bit. (In case my question is ridiculous, I'm a student who has no experience dealing with zeta functions - I've only ever dealt with their components.)
I was wondering if it would be possible that a lot of luck/creativity would allow one to simplify $\zeta(s)=0$, or an analytic continuation of $\zeta(s)$, and show that $\Re(s)=1/2$? Or would a proof have to involve some more abstract, qualitative ideas? If the latter, is it likely that there are (undiscovered) non-trivial ideas that would just require a really creative simplification?
I mean, I'm playing around with it because it's fun/good practice...but I'm still curious if there's a remote possibility that an amateur (i.e., someone with a math degree and some graduate courses under their belt) could find something.
Thanks for any feedback.