Dirichlet L-functions and primes in progressions

Dirichlet characters and L-functions, from Dirichlet's theorem to average distribution and the least prime in a progression. The course develops conductors, Gauss sums, character sums, functional equations, special values, zero-free regions, exceptional zeros, explicit formulas, the prime number theorem for progressions, Siegel and Siegel–Walfisz estimates, the large sieve, Vaughan's identity, the Bombieri–Vinogradov and Barban–Davenport–Halberstam theorems, and Linnik's theorem. The opening lessons prove the character constructions, quadratic reciprocity and elementary analytic prerequisites used in their arguments.

1. Dirichlet characters
2. Dirichlet's theorem on primes in arithmetic progressions
3. Gauss sums
4. Character sums: the Pólya–Vinogradov inequality
5. The functional equation of Dirichlet L-functions
6. Values of Dirichlet L-functions at s = 1
7. Zero-free regions and the exceptional zero
8. Counting zeros and the explicit formula for ψ(x, χ)
9. The prime number theorem for arithmetic progressions
10. Siegel's theorem
11. The Siegel–Walfisz theorem
12. The large sieve inequality
13. The multiplicative large sieve and bilinear forms with characters
14. Vaughan's identity and sums over primes
15. The Bombieri–Vinogradov theorem
16. Mean square distribution: the Barban–Davenport–Halberstam theorem
17. The least prime in a progression: Linnik's theorem

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