Primes in Arithmetic Progression

This lecture delves into the concept of primes in arithmetic progression, exploring the relationship between Dirichlet L-functions and characters. The instructor discusses the orthogonality of characters, the analytic continuation of L-functions, and the class number formula. By analyzing the behavior of L-functions at s=1, the lecture aims to demonstrate the divergence of the sum of 1 over p for p congruent to a modulo q. The proof involves showing that the L-function for non-trivial characters tends to infinity as s approaches 1. The lecture also touches upon Legendre symbols and their connection to real characters. Through a detailed analysis, the instructor illustrates the intricate interplay between characters, L-functions, and prime numbers.