How to approach experts to get feedback on a paper I'm a PhD student in pure maths at a university in Europe. For reasons that I won't go into, my research interests do not coincide with my supervisor's ones. This means that, although she gives me good general advice, she can't give me more "technical" feedback on my work. I've now written a complete first draft of my first paper, and I'd like to get some external feedback before thinking of submitting it to a journal. My question is:


  What is the best way to approach the experts in my field (for whom I am a complete stranger) to politely ask if they can read my paper and give some feedback on it?


Emailing, of course, would be the preferred means of communication, so suggestions on how to structure a potential email to send to the experts are very welcomed.
Generally it's considered okay to e-mail researchers who you don't know and share your work with them. This is especially true if you are a student.

Be conservative about it -- pick only a few people whose research interests align closely with the paper you're writing. (For example, whose work are you citing?) Don't e-mail famous people just because they're famous.

That said, I think that asking for detailed feedback might be seen as presumptuous. Rather, just tell them that you would like to share your work with them, describe it very briefly, and say something like "If you have any comments I would be grateful to hear them."

Probably most people will ignore your message. (They have no obligation whatsoever to read your message or respond.) Maybe one or two will write back with general feedback or suggestions, and if you are very lucky you might get some detailed feedback. What you might realistically hope for is a response along the lines of "Thanks, this looks interesting, you might be interested in X, Y, and Z" which at least gives you some related works to check out.