The Bateman-Horn Conjecture: Heuristics, History, and Applications

The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.Comment: 47 pages, formerly titled "One conjecture to rule them all: Bateman-Horn," to appear in Expositiones Mathematica