Some Fascinating Characters in Number Theory

Are there infinitely many primes of the form n^2+1, where n is an integer? No one knows. In fact, there’s no example of any (single variable) polynomial of degree 2 or greater that’s been proved to output infinitely many primes. By contrast, the linear polynomial n+1 outputs infinitely many primes, a fact that’s been known for over 2000 years. Rather less trivially, Dirichlet proved in 1837 that any linear polynomial of the form an+b with a, b coprime must output infinitely many primes. To make his proof work, Dirichlet introduced certain nice functions called characters, which evolved (over the course of the next hundred years) into fundamental objects of study in algebra and number theory. I will discuss some of the history and mathematics of Dirichlet’s characters, including a very recent and simple characterization of them that seems to have been previously overlooked.