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

Видео Zeta Explained #87: Prime Progressions and L-Functions (Part 2) канала Zeta Explained