How to get a .edu email address as an independent mathematician? I am a hobbyist mathematician in China. I study maths by myself and make some course videos to teach commutative algebra, functional analysis and other topics on the internet, like the Khan Academy and MIT open courses. 

Now I need a .edu email to get into some sites such as ResearchGate and arxiv. Is there any organisation that will help me like this: I show them some material, such as videos and papers, and if they think I am no weaker than some college teachers at least, they would give me a .edu email?
Of course, becoming a student or staff member is one way to get a university email.
Some institutions provide alumni email.

Many universities have various unpaid affiliates. Such affiliates are sometimes eligible for a  university email. However, such affiliates are often expected to contribute to the school, faculty, or university. For example, you might publish with the university as your affiliation or you might give occasional lectures or you might supervise a research student. These sorts of affiliations are typically obtained by building up a relationship with some academics in a given department and making enquiries.