What are the subjects an analytic number theorist must be well versed with after undergraduate studies?

Question

I am a mathematics major and I aspire to be an analytic number theorist. In general, what are the subjects an analytic number theorist must be well versed with after undergraduate studies (i.e. in masters level)?

Answer 1

Here's what you must know in order to start learning analytic number theory:

• complex analysis
• elementary number theory

You won't be able to get very far into a text like Montgomery & Vaughan or Davenport without a solid background in both of the above.

Here's what you might want to know to help you learn more advanced areas of analytic number theory:

• Algebraic number theory, especially pertaining to quadratic extensions of $\mathbb{Q}$ (for problems about quadratic characters, fundamental discriminants, etc.)
• Quadratic forms
• Representation theory (mostly for things related to modular forms/automorphic representations)
• Geometry of Lie groups, especially of $\mathrm{SL}_2(\mathbb{R})$ (again for modular forms and related topics)
• Basic ergodic theory and dynamical systems (for problems related to, say, Sarnak's conjecture on the disjointness of the Möbius function to zero entropy flows, or quantum unique ergodicity on the upper half-plane)

Then there are specialised tools/areas of number theory that are handy to know about, like sieve theory, or the Hardy-Littlewood circle method. I'd say those are very specific though.