What are the allowed axioms to solve the Riemann Hypothesis? First, is it even possible to state a well known problem and know what axioms are allowed for it? I'd guess that one must look at the branches of math involved, but I'm no mathematician.
Therefore, I would like to ask the people working on this problem if this there is a fixed list of allowed axioms for this problem.

Most of everyday mathematics works with a set of axioms known as ZFC. Some mathematicians study possible (small) changes to that axiom system, with various interesting (to them) implications. But most of the alternatives proposed do not affect work on most open problems.
Here's an analogy. There are (probably) engineers who spend time thinking about ways to formulate concrete to make it a little cheaper, or a little stronger, or a little longer lasting. But what they do does not affect the design of skyscrapers, which will be stable with foundations built with the current industry standard concrete, and would be with any of the known acceptable variants. So architects don't really have to pay attention to the concrete.
Rarely. from time to time, there's a building design challenge where the construction does depend on those variants in the foundation.
I think most mathematicians think that the Riemann hypothesis is analogous to designing a really tall skyscraper. New construction techniques will be required, but new foundations won't. 
