Zeta Explained Course

Currently Playing: Zeta Explained #86: Prime Progressions and L-Functions (Part 1)

This is the 86th 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 covers Dirichlet's Theorem on primes in arithmetic progressions. Instead of proving the general theorem (which will be done over the next 2 videos), we prove the specific case for q=6, i.e. that the sequences 6k+1 and 6k+5 each contain infinitely many primes. Our proof will follow Dirichlet's ideas, so we will encounter Dirichlet characters and Dirichlet L-functions. Though, we will take shortcuts at certain points due to how nicely 6 behaves, avoiding the complexity of Dirichlet's general proof until the next 2 videos. 00:00 - Sieve of Eratosthenes and primes 07:16 - Euler's proof of infinitely many primes 13:14 - Modular arithmetic on 6 16:40 - Dirichlet L-functions 18:44 - Linear combination of L-functions as a filter 22:25 - Generalized Euler product formula 25:20 - log L(1,χ_0) = ∞ 27:55 - |log L(1,χ_1)|＜∞ 31:51 - Filter to primes 33:59 - End of proof 36:56 - Outro