You are free to do a project about any "non-trivial" subject related to what we discussed in class. That's pretty vague, making it hard to pick something, but it's also pretty open ended, leaving a lot of choices. Here are a few ideas for a project:
- 1- Selberg's Elementary PNT Proof: The proof of the PNT we're doing in class relies on some analysis (the Tauberian Theorem) to reach its conclusion, and for quite some time, people thought that that was the only way you could prove the PNT, but in the 1950s, Atle Selberg, with a little help from Paul Erdos, came up with a proof for the PNT without any analysis at all. You could explain his proof.
- 2- Interdisciplinary Proof Attempts: Normally, a conjecture only receives attention from the field that started it, but the Riemann Hypothesis has received attention from more than just analytic number theorists. There have been proof attempts from Operator Theory, Non-commutative Geometry, and much more. You could look at some of these different methods.
- 3- Numerical Analysis of the Zeros: Finding zero-free regions of the critical strip and calculating the imaginary parts of the zeros of the zeta function are of great importance in number theory, so the algorithms that do these things have become quite advanced. If you have a background in algorithmic analysis, this direction might interest you.