Primes, Arithmetic Functions, and the Zeta Function

Abstract

In this chapter we will discuss properties of primes and prime decomposition in the ring A = F[T]. Much of this discussion will be facilitated by the use of the zeta function associated to A. This zeta function is an analogue of the classical zeta function which was first introduced by L. Euler and whose study was immeasurably enriched by the contributions of B. Riemann. In the case of polynomial rings the zeta function is a much simpler object and its use rapidly leads to a sharp version of the prime number theorem for polynomials without the need for any complicated analytic investigations. Later we will see that this situation is a bit deceptive. When we investigate arithmetic in more general function fields than F(T), the corresponding zeta function will turn out to be a much more subtle invariant.