Introduction to number theory lecture 49. Dirichlet's theorem

This lecture, part of a Berkeley undergraduate number theory course, provides an overview of the proof of Dirichlet’s theorem, which states that there are infinitely many primes in any arithmetic progression a, a+n, a+2n, … where a and n are coprime. The lecturer begins by recalling special cases previously proved, such as primes of the form 4n+1 or 3n+1, and notes that these elementary methods fail in general. He then introduces the key idea: generalizing Euler’s proof of the infinitude of primes using the Riemann zeta function. For each modulus n, one defines Dirichlet characters, which are multiplicative functions periodic modulo n that vanish on numbers not coprime to n. The associated Dirichlet L-series are then defined. The proof of Dirichlet’s theorem reduces to showing that these L-series do not vanish at s=1. The lecturer illustrates the construction of Dirichlet characters for several small moduli (1,2,3,4,5,6,7,8,12), showing how they arise from the group of units modulo n. He explains that for prime moduli, characters are given by powers of a primitive root, while for composite moduli like 8 and 12, they are built from characters of prime power factors via the Chinese remainder theorem. The lecture concludes by stating that the number of Dirichlet characters modulo n equals phi(n), and that the next lectures will cover further properties and the deduction of the theorem from the non-vanishing of L-series.