Zeta Explained Course

Currently Playing: Zeta Explained #87: Prime Progressions and L-Functions (Part 2)

This is the 87th 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 86), we proved the specific case for q=6, i.e. that the sequences 6k+1 and 6k+5 each contain infinitely many primes. However, we set q=7 for this episode and see that things get much more complicated. We examine the Dirichlet characters and linear combinations for q=7. 00:00 - Primes in arithmetic progression 02:06 - Modular arithmetic on 6 and 7 07:50 - Dirichlet characters for 6 and 7 10:33 - Visualization of Dirichlet's idea for 6 and 7 14:09 - Dirichlet L-functions and linear combinations 18:42 - Proof of general linear combination equation 23:49 - Filtering L-functions to primes 25:40 - Final linear combinations