Representation theory: Dirichlet's theorem
In this talk we see how to use characters of finite abelian groups to prove Dirichlet's theorem that there are infinitely many primes in certain arithmetic progressions. We first recall Euler's proof that there are infinitely many primes, which is the simplest case of Dirichlet's proof. Then we define Dirichlet characters and Dircihlet L-functions and give some examples. We prove Dirchlets theorem for the special case of primes that are 3 mod 8 by using the orthogonality relations of Dircihlet characters, together with the fact that Dirichlet L-series L(s,X) do not vanish at s=1. Finally we sketch how to use the orthogonality relations to show the non-vanishing of Dirichlet characters at s=1.
▶︎
Representation theory: Orthogonality relations
▶︎
Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations
▶︎
Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem
▶︎
Introduction to analytic number theory 2: Dirichlet characters and Dirichlet L-functions.
▶︎
Representation theory: Introduction
▶︎
The Key to the Riemann Hypothesis - Numberphile
▶︎
Irreducible representations (irreps) | Representation theory episode 3
▶︎
William Dunham, A tribute to Euler
▶︎
The Dirichlet Integral is destroyed by Feynman's Trick
▶︎
John Thompson: Dirichlet series and SL(2,Z)
▶︎
Introduction to number theory lecture 49. Dirichlet's theorem
▶︎
L-functions and the Langlands program (RH Saga S1E2)
▶︎
The distribution of primes and zeros of Riemann's Zeta function - James Maynard
▶︎
Representation theory: Abelian groups
▶︎
Table of Dirichlet Characters mod16
▶︎
Judge Can’t Stop Laughing At Sovereign Citizen’s Courtroom Meltdown!!!
▶︎
Ecuador vs. Germany Highlights FIFA World Cup 2026 | Sportschau
▶︎
Representations of Finite Groups | Definitions and simple examples.
▶︎
Math313 Lec09 MotivationDirichletThm
▶︎