Summer Internship - A very well-known Professor I want to send an e-mail to a very well-known professor in the field of my interest. I have read most of his papers and I am 90% sure that I like to work under his supervision at least for some months. I am an undergraduate student but being realistic, I hope to be considered a good student in my field and recently, I published a paper in JHEP. My question is:


  The professor has a research group in a French university. I want to let him know I will pay for my costs and I just want to do a 3-month or even more research under his supervision. Since I guess I have good accomplishments in my CV, how I can ask him politely for a summer internship (or a visit, I am not familiar with these names) and potentially guarantee to receive a reply message from him?


Thanks!
I want to let him know I will pay for my costs


This is not how things are supposed to work. It is normal that young researchers visit other institutions, but these visits are normally paid either by their current institution, or by the accepting one. What you're doing is explicitly proposing to do unpaid work, which is a bad practice and you don't want to work with someone who agrees to that.

What you can do instead is find 5-10 places where you'd like to go for an 'internship' or 'placement', etc and send emails with inquiries. It is fine to ask a professor whether they can hire you as an intern, but I don't think you should propose unreasonable arrangements.