Naively, the answer is "any good ol' axiomatic system which is not obviously inconsistent and can provide a sound basis for arithmetic".
So you can work with Peano Arithmetic, or the second-order Peano axioms, or one of many set theories, or just $\sf ZFC$, or augment $\sf ZFC$ by any large cardinal axiom, or by axioms like $V=L$, and much more.
The thing here, is that almost none of this matters. The Riemann Hypothesis is equivalent to a $\Sigma_1$ sentence in the language of arithmetic, let's call it $\varphi$ for now. This is an almost irrelevant fact, but it has an interesting consequence: it is enough to prove that $\varphi$ holds for the natural numbers in order to prove the hypothesis holds.
And it almost doesn't matter what is the axiomatic system you're using, since most of the basic axiomatic systems that allow development of arithmetic will agree on whether or not a $\Sigma_1$ sentence is true in $\Bbb N$.
(There is a caveat here, that these systems need to agree on what is $\Bbb N$ to begin with, but let's ignore this for now, because we can, and even a proof under this assumption would be extraordinary and probably accepted as a proof by the bulk of mathematicians.)