Is it appropriate to email a professor asking for a possible summer project, through online correspondence? I'm an international Masters student, aiming for a certain reasonably competitive pure math PhD program in the US next year. In Dan C's answer here, he suggests to try and get a recommendation letter from one of the faculty in the school you apply to, by doing some work under him/her. There are a couple of professors in the program whose work interest me, and one of whom I might select as my advisor (given that I'm selected for the program, of course. And assuming my interests don't shift by the next year). However, since I'm not from the US, it would not be possible for me to meet the professor in person.

Given this premise, is it appropriate to contact the professor asking for a possible summer project through online correspondence? Also, should I mention that I would be applying to the program next year?
Given this premise, is it appropriate to contact the professor asking for a possible summer project through online correspondence? 


I doubt you'll get a positive answer, but there's nothing offensive about asking, as long as the way you phrase your e-mail doesn't come across as entitled or demanding.  Given that you are concerned about whether it's even appropriate to ask, I'm not too worried about that.

There are a number of reasons why this probably won't work out: supervising a student can be a fair amount of work, online correspondence is an inefficient way to do it, and there's usually plenty of demand from local students already.  However, it can't hurt to ask, and you might happen to ask just when the professor has in mind a project that none of the local students have the time or expertise to take up.


  Also, should I mention that I would be applying to the program next year?


I think you should.  It's a tricky issue, since you don't want to make it sound like that's the only reason you're interested (and learning or doing research are secondary), or like you expect that it will lead to guaranteed admission.  In fact, I'd recommend explicitly addressing these issues, perhaps by including something like this partway through your e-mail: "For context, I should point out that I'm planning to apply to the [university] Ph.D. program next year, and you are someone I have in mind as a possible advisor.  Of course I understand that I may not be admitted, but working on a summer project would be a valuable learning experience regardless of where I end up attending graduate school."