Diversifying the academic institutions one attends I am planning on doing research mathematics at a university or similar institution as a career. I have heard that it is important to diversify your studies (not attend the same school for undergrad, masters, and PhD). Is this true? What if I am already going to a good school, with the potential to work with people in the forefront of my field?

I am currently doing my undergrad degree, but when it comes time to do my masters, is it important to find a different school, or will staying in the same place hurt my job prospects? Any advice would be greatly appreciated!
While this is generally true, it depends heavily on the field and subfield. Pure math is a little unique compared to many other disciplines, and in my experience has a much higher rate of students staying with the same university/department/advisor from undergrad through PhD--perhaps because if you find a professor working on a given topic you'd like to study, there's a decent chance he or she is the only (or one of very few) person in the country doing it. I know a decent number of people in pure math who did this, and at least two of them are now tenured math professors at research universities--so I don't think it's guaranteed to hurt your job prospects. 

I would talk to the professor(s) you're interested in working with and ask for suggestions specific to studying analytic number theory.