Zeta Explained Course

Currently Playing: Zeta Explained #88: Prime Progressions and L-Functions (Part 3)

This is the 88th video in a series explaining the Riemann zeta function. The idea of the series is to start with basics and eventually work our way to advanced topics. We loosely follow Davenport's "Multiplicative Number Theory". This particular video is covers Dirichlet's Theorem on primes in arithmetic progressions. Last time (Ep 87), we started the proof for q=7, i.e. that the sequences 7k+1, 7k+2, 7k+3, 7k+4, 7k+5, 7k+6 each contain infinitely many primes, and proved a linear combination formula to isolate prime sequences. In this video, we complete the proof by showing that the log of the L-function are either infinite (for χ_0) or bounded (for χ ≠ χ_0). 00:00 - Intro 05:06 - Roadmap 06:50 - Case 1: Proof that L(1, χ_0) = ∞ 09:38 - Case 2: Proof that for χ ≠ χ_0, L(1, χ_0) is finite 12:20 - Case 3A: Proof that for χ ≠ χ_0, χ complex, L(1, χ_0) ≠ 0 20:41 - Case 3B: Proof that for χ ≠ χ_0, χ real, L(1, χ_0) ≠ 0 31:34 - Summary