Chapter 4

Prime numbers in residue classes: Dirichlet's theorem

The number $30n + r$ is a multiple of 2,3 or 5 unless $r$ is one of: 1, 7, 11, 13, 17, 19, 23, 29. So all prime numbers (apart from 2,3,5) are in one of the eight ``residue classes" $\{ 30n+r : n \in \mathbb{Z} \}$ corresponding to these values of $r$. Do they prefer some classes to others? Among numbers from 1 to 3600 (i.e. 120 blocks of 30 consecutive numbers), the primes are distributed between the classes as follows:

$$\begin{array}{cccccccc} 1 & 7 & 11 & 13 & 17 & 19 & 23 & 29 \\ 62 & 63 & 62 & 62 & 64 & 60 & 62 & 65 \end{array}$$

The table in Appendix G displays the primes up to 2520, classified in this way. The distribution appears to be very nearly equal - but is this a hand-picked example?

Of course, there is nothing special about 30: it can be replaced by a general integer $k$, and the possible classes are then represented by the numbers $r$ such that $(r,k) = 1$. In this chapter, we shall prove a famous theorem stating that, asymptotically, the prime numbers are indeed equally distributed between these classes. The theorem was essentially discovered by P.G. Dirichlet in 1837, no less than 59 years before the proof of the prime number theorem itself. Dirichlet actually proved that the series $\sum (1/p)$, taken over the primes in any particular residue class, diverges (so that there are infinitely many such primes), but when his ideas are combined with those used to prove the prime number theorem, they deliver the result stated.

The method depends on a branch of mathematics that we have not used so far, namely group theory. For a fixed $k$, the residue classes of numbers coprime to $k$ form a group. This is how group theory enters into number theory, and it provides exactly the right concept to reflect the symmetry between these classes. We only need quite basic group theory, and only abelian groups. The key concept for Dirichlet's method is that of characters of abelian groups, which we describe in section 4.1, assuming no previous knowledge. Dirichlet characters, for each particular $k$, are essentially the characters of the group just mentioned. For us, they form an interesting new class of arithmetic functions. The corresponding Dirichlet series are called $L$-functions. The desired theorem is then obtained as another case of our fundamental theorem, with the $L$-functions playing a part analogous to the role of the zeta function in the proof of the ordinary prime number theorem.

Dirichlet's theorem, and the route to it, amount to a highly elegant and worthwhile extension of the proof of the prime number theorem. The reader is about to enter an especially beautiful area of the mathematical landscape!