Are you currently taking students?
Maybe. Here are some reasons I might not be: I have too many students now, I'm about to take a job somewhere else, I'm about to retire, I'm already dead. I probably am, but it's prudent to ask.
[if applicable] I want to work in X. Would you be able to advise me on this topic?
That depends on what X is. If it falls within the realm of my research areas, then yes. If it's closely related, then maybe. Bear in mind that the closer your topic is to my work, the better I will be able to advise you (I'll know more about what's been done, have better ideas for how to approach your problem, and be better able to introduce you to experts in your field). Also, be aware that you may change your mind about what you want to do when you learn more, and that you can change your research topics later in your career. This is true even for experienced researchers.
What's your research about?
Most of my work is in number theory, automorphic forms and representation theory. Within this area, the bulk of my work focuses elliptic curves, modular forms and L-functions. I particularly like investigating algebraic structures in arithmetic problems and understanding links between different types of structures. I also do some combinatorics/graph theory. I will tell you more when you ask in my office, but I won't be able to describe all aspects of my research in one go.
What should I do if I'm interested in working with you?
Like other members of the lepus genus, I will get startled and run away if you ask me to be your advisor too suddenly. At some point, if you're seriously considering me as a potential advisor, come talk to me about this and we should arrange to do a reading course together. This will give both of us a chance to get some sense of what it would be like to work together. Even if you've taken a graduate course from me, I'll generally want you to do a reading course with me before considering being your advisor. You should do your best to impress me in the reading course (see below).
What do you expect from me in a reading course?
For any course and student, I expect at least 3 hours of work per week for each credit hour. In this case, regardless of how many credit hours you sign up for, I'll usually want a commitment of on average at least 10 hours/week to make sure you get a good start towards your dissertation, and have a decent sense of what you're getting into. Remember, you should be trying to impress me and, if you work with me, this material will likely be crucial for your thesis, so I expect you to give this reading class top priority. (If you can put in more time, that's great! You'll need to when you move to the research phase anyway.)
Ideally, we should discuss the goals and plan of the course before the start of that semester. I'll want to meet every week or so to check on your progress and address any questions you have. To make sure you're really learning the material, I will probably have you do one or more of the following: (i) do homework exercises, (ii) have you (informally) present to me material you read and/or your homework solutions at the board, (iii) write up notes on the material and/or your homework solutions.
So work hard, but there's no need to get worked up about our meetings. While this is sort of a test to see if I think you can do a PhD under me, I'm not expecting genius or perfection. It's good to feel some pressure and have a sense of urgency, but not to the point where you're uncomfortable around me or discouraged about your studies. The things I want to see are that (1) you're working hard, (2) you're motivated and (3) you're making progress. I don't care if you're brighter or duller than me, if you make lots of mistakes or if you need me to explain things to you 3 times before it makes sense. Except for possibly a rare few, we've all been in that stage (and some of us still are). What's important to me is your attitude and effort (you of course also need to be reasonably competent, but you probably are if you've gotten this far.)
In summary, work hard and relax. If you're not sure if you're doing enough, ask me. And if you're not, I will tell you.
How often should we meet?
Some advisors are more hands-on, and some are more hands-off. I usually like to meet with my students every week or so to make sure you're making progress and help keep you focused. (This answer applies to both reading courses and the research phase.)
Will you be my advisor?
Maybe. Some people find it hard to say no to this question without a solid reason like the one I gave above, but I'm perfectly happy to say no if I'm not convinced of your commitment, work ethic and ability. Advising students is a very time consuming task, and I have had previous experiences with students who started to work with me but seemed to lack the motivation and/or diligence required to do a PhD in math. (This is not to say I would view you as a burden--I enjoy working with motivated students.)
What would you expect of me as a student?
Mostly the same as what I expect of you for a reading course, but more. Once you're officially working with me, I expect you to put in at least 20 hours/week for your research (this includes time spent learning relevant things and time spent on relevant courses). This includes summer and winter breaks, though it's okay to take it easy for a week or two. Of course you should not view the 20 hours/week as the goal, but merely a benchmark of the minimum amount of effort I expect and a tool to help keep you on track. You will need to put forth a great amount of effort to learn the background for your thesis project, let alone doing something new, and at certain stages you may need to work harder than others. It's natural to get stuck and discouraged at various stages. Even if you are making some sort of progress (and you are as long as you're learning something), it sometimes may not feel this way. This "20 hour rule" is one way to help you get through these rough patches.
I'll also expect you to keep me abreast of your progress on a regular (usually weekly) basis. As long as we're meeting often, I will do what I can to help make it easier for you when the going gets hard. Remember, the more effort you put in, the more effort I am willing to put in to help you.
What do I need to learn to start working on a thesis problem?
This will depend a lot on what your thesis topic will be. Some advisors may have standard prerequisite reading for research, but I don't. You will probably have to do a year or so of specialized reading before moving on to the "research stage," i.e., until you have a potential thesis problem. In any case, there is no clear line between reading and research anyway--probably at least half of actual "research" just consists of understanding what has been done so far.
What do you want to know about me before making a decision and/or to find a thesis topic for me?
As I said before, the main things I want to be find out about are your dedication, work ethic, and your competency, which I should get a sense of from a reading course. However, there are some other things I want to know: what you've already learned, what direction you'd like to go (if you have an idea), and what you're thinking about after your PhD (again, if you have an idea). All of these are important factors in helping you find an appropriate thesis topic, and consequently for directing your reading courses before you start getting into actual research. As you refine your ideas about these, please keep me updated. (Ideally, if you do a good thesis, you should be able to get any kind of job: teaching, research or industry. But in practice, certain theses will make it easier for you to get specific kinds of jobs, or jobs in certain geographical regions.)
Should I ask you to be my advisor?
If you like the kind of math I do and felt comfortable working with me in a reading course, then sure. What I want to emphasize is that, apart from special circumstances, there are two main things you should consider: (i) that you can study a topic you like, and (ii) that you get along well with me. In my opinion, both of these are of roughly equal importance (assuming the advisors you're debating between are similarly research active). Many students seem to choose an advisor solely on what they think they want to research, but in fact there are probably lots of different areas you could be happy working in (and you may change areas later anyway).
Do you have a problem for me?
Maybe. Eventually I should, and this is a reasonable question to ask at some point. (While shopping for advisors, it's fine to ask if they usually suggest problems to students or expect them to find their own, or what kind of problems their students have worked on. But good problems are hard to find, so asking someone to actually give you a potential thesis problem before you've agreed to work together is perhaps premature.) I typically have a couple of problems (at least vaguely) in mind for suitable students (depending on their interests, background, and capabilities), though I may need to think quite a bit to suggest an appropriate problem specifically for you. In any case, I probably won't expect you to come up with a problem entirely on your own, like some advisors do.
How much will you help me with my thesis?
A lot. I will help you choose background reading, suggest a topic/problem, and probably suggest one or more approaches. I will try to help you when you get stuck, and will try to set incremental tasks so you can see that you are making progress. I will also give you a lot of suggestions and feedback with the write-up. However, I will not solve your thesis problem for you (or write it for you).
What else will you do for me?
I will suggest conferences/workshops for you to go to, introduce you to people, tell people about your work, and generally give you career advice. Depending on how much grant money I have, I can help support your travel and/or get you out of some teaching duties. When the time comes, I will help you with your job applications, and generally support your career after you graduate. For instance, I am willing to give you suggestions/feedback for writing/submitting papers or grant applications, can probably give you suggestions for research problems after your thesis, and write recommendation letters for you.
How much do I need to do for my thesis?
Roughly you should have done enough "new" stuff to get 1-2 publishable papers out of it. Of course, as a student you may not have a good sense of what is "publishable," but that's what I'm here for, and I'll tell you specific goals for your thesis. You may be worried about your ability to do something "new," but don't worry, your thesis doesn't have to have groundbreaking new ideas--the majority of theses out there boil down to working out details of ideas already known to experts. (That doesn't mean these theses are bad--many of them are very good. Details can be very important.) Hopefully you will be able to contribute some new ideas, even minor ones, but a large part of your time will be spent just learning the necessary math to carry out your project. Of course, things often don't work out as planned, so you may not meet your original goals (I didn't for my thesis), but in my experience, with sufficient time and effort, you will get something new. Be patient, and keep at it. In any case, the primary goal of a PhD thesis is not to blow peoples' minds, but to train you in research.