Primes in Arithmetic Progressions

The plan for the proof of Dirichlet’s theorem is given before embarking on the proof itself. There follow descriptions of the prime number theorem in arithmetic progressions, the generalised Riemann hypothesis, the Siegel–Walfisz theorem, Linnik’s theorem, the Brun–Titchmarsh theorem, Bombieri’s theorem, and the Titchmarsh–Linnik theorem. The very difficult problems concerning Chebyshev’s bias, or ‘prime number races’, are considered, including the Rubinstein–Sarnark hypothetical theorem. The chapter ends with the notion of primitive characters and the Pólya–Vinogradov inequality.