On primes in arithmetic progressions

a, a + q, a + 2q, a + 3q, . . . in which a and q have no common factor and q is prime. The general case, for arbitrary q, was completed only later by him, in 1840, when he had finished proving his celebrated class number formula. In fact, many are of the view that the subject of analytic number theory begins with these two papers. It is also accurate to say that character theory of finite abelian groups begins here. In this chapter we will derive Dirichlet’s theorem, not exactly following his approach, but at least initially tracing his inspiration.