Let me preface this by saying this answer will almost certainly be obsolete within months, and so perhaps MO (which is supposed to be for more definitive answers) is not a good fit for your question. I have only used Gemini and ChatGPT/Codex, because those are what my university subscribes to. Our ChatGPT subscription is very good with a large budget, while the Gemini subscription has a tiny budget before reverting to the free version. Unfortunately, there's a enormous difference between free and older AI and ChatGPT 5.5 and especially 5.6 Sol. (My second-hand understanding is that modern Claude is more comparable to the latter category.) Most notably, old and cheap AI will give you a good answer if it can, but will always lie and try to trick you rather than giving up. This makes it of somewhat limited use beyond some specific use cases.
Here are some of the main things I think free AI is useful for:
- Making Tikz diagrams. (You can just upload a hand-drawn file, ask it to produce Tikz code, tell it what it got wrong, and then you're usually done.)
- Debugging any code, especially LaTeX (which has useless error messages). Just drop in your LaTeX file, and it should be able to tell you where the error is.
- Checking over assignments for errors or misstatements.
- Checking papers for errors.
Any of these can be done with any of the main free AI programs such as ChatGPT, Gemini, or Claude. Note that none of these are actual research mathematics.
ChatGPT 5.6 Sol is actually good at doing mathematics (and some of this is true of 5.5 or 5.6 Luna to a lesser extent). For the first time, it seems to actually care about getting things correct rather than bullshitting you. Unfortunately, it's expensive. But it can basically just do any routine calculation you want (even if that requires some moderately complicated software), and if you give it a clean statement and that statement can be proved routinely in less than 10 pages, it will find the proof. The main caveat to this is that it's worse at anything visual than things that are purely symbolic. In particular, modern AI will quickly solve any IMO-style question, any prelim-level problem, and any graduate-level homework problem. What's especially useful is that it can prove things that are "routine" to other mathematicians, even if they're not routine to you. You can also say "I think I can prove X using idea Y" and it'll give a quick check as to whether you missed a nuance or an additional hypothesis.
Sol can still make mistakes, but unlike earlier models, the mistakes tend to be recognizable human mistakes that a real collaborator might have made (missing a technical assumption, relying on a false statement in the literature, etc.) You sometimes have to be a bit careful reading what it's written because it will sometimes look like it's answering your question, but if you read carefully it's only said: "If you assume X (which I haven't proved yet), then Y would follow." It's also finally good at searching the literature. I find it especially helpful for stuff where you remember a few vague things about some result or paper and want to find the paper.
Unfortunately, it is garbage at explaining arguments. The explanations are a bit better if you have it produce LaTeX rather than just respond in the chat, but they're still terrible. Badly structured. No sense of where the critical steps are. Makes up new words without defining them. Skips over difficult details. It almost always takes a bit of a conversation, and some time away thinking yourself, to gain any understanding (which is already true of reading math papers, but this is worse than all but the very worst written math papers). It's reminiscent of talking to a very clever teenager who hasn't yet learned to explain anything. It's much better at explaining examples than arguments, so asking it to explain the argument in an example can be very useful.
I don't think there's currently a great way to see what cutting-edge AI is capable of at a genuinely low cost. ChatGPT did just drastically lower the cost of its faster/cheaper 5.6 "Luna" model. If you want to get a good taste of relatively modern AI without spending a lot of money, I'd suggest Luna at High (or Extra High) intensity on a $20/month subscription, which should be a reasonably high budget that lets you play around a lot. (There are, sadly, obvious and substantial moral problems with giving $20 to OpenAI, but that's a whole separate question. My university subscription was purchased without my input, and also has much more data protection than a paid account.) Use the "work" mode rather than "chat" mode so that it can produce files for you to read and run code.