During a PhD you are normally expected to conduct a novel research with the help of an advisor. This research can be about almost any problem in the area you chose to obtain a PhD in. A PhD in pure math normally involves proving or getting closer to a proof of some important conjecture (which doesn't necessarily mean trying to prove one of the notoriously hard ones), or even giving a novel proof to an existing theorem. A PhD in applied math and in computer science will be something more rooted into solving a practical problem, like developing an algorithm that performs a given task in a more efficient way, etc. Proving a theory wrong would be worth a PhD in the field the theory comes from (as most theories are built to be self-consistent said proof would involve performing experiments and showing the theoretical predictions don't hold).
