How do I apply for a PhD position via email? Should I read papers written by prospective advisors beforehand and write about it? I am a master degree student. I am keen to join a PhD program under a mathematician in a university in France or Luxembourg. I found that in the webpage it is written that I should send an email with my research interests to the faculty. I have some questions regarding those. My questions are as follows:

Do I need to read research papers of the faculty before sending the email and in the email should I include that I have read his/her paper (or some parts of it)?
How should I write the email?
You should look at some of the professor's recent papers so that you can get a sense of what they're currently working on, and whether it is of interest to you. However, I do not think it is necessary to refer to specific papers when you first email the professor. Far too often when people do this, they end up giving a negative impression: it rapidly becomes clear that they haven't actually read the paper in detail, or have misunderstood its content.
Keep it brief: a sentence or two on who you are (what/where you've studied), a sentence or two on what you're asking for (Their suggestions for a project? Funding? Are you hoping to start immediately, or at some point in the future?), and a sentence or two on the general research area(s) that interest you. If you have already written a masters thesis or similar, put it online somewhere and include a link.