I am finishing my Master's in mathematics in Germany and I'd like to apply for a PhD in Europe, preferably in the UK. Most departments recommend that students get in touch with potential supervisors prior to submitting a formal application. I am a little nervous about it and I would appreciate advice regarding the following:
(1) Some people say it's advantageous to mention interest in specific papers published by a given professor. But I'm not sure how applicable this is to mathematics. To be honest, I haven't read a single paper by most of the people I'd like to apply to. (Reading and understanding a math paper takes a long time, so I think it's rather normal.) Is it OK just to say, for example, "I've seen you have published a lot of papers on non-linear PDEs, which is an area I'd like to do research in", or does this sound too generic?
(2) Is it OK to mention that my Master thesis supervisor or lecturer at my university recommended a given professor to me as a potential PhD advisor (they know each other), or does this sound somewhat awkward/patronising?
(3) How long should my email be? Is about 300 words too long?
(4) What should a first email accomplish? Should I just introduce myself and express interest? Or should I ask some specific questions about a potential research project straightaway?
I will really appreciate your advice, especially from academic mathematicians. I think one of the problems is that I find it a bit hard to see how the situation looks from the perspective of the potential supervisor. Do they get hundreds of such emails every year and just get annoyed when they get another one? Do they want the applicants to be very specific from the start, or is it better to first introduce oneself and see if they are at all interested before asking more specific questions about a research project, etc.?