prime number theorem Goldbach conjecture twin prime conjecture Hardy–Littlewood conjectures Waring problem Riemann hypothesis circle method sieve methods L-functions modular forms automorphic forms analytic number theory Goldbach
The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.
