Birthday of Dirichlet (1805) — Primes in Every Arithmetic Progression

Peter Gustav Lejeune Dirichlet was born in Düren on February 13, 1805. He proved in 1837 one of the most beautiful theorems in number theory: if a and b have no common factor, then the arithmetic progression a, a+b, a+2b, a+3b, … contains infinitely many primes. So the sequence 7, 17, 27, 37, 47, 57, 67, … contains infinitely many primes. The proof required he invent what we now call Dirichlet series and L-functions — tools that imported complex analysis into number theory for the first time. He also formalised the modern definition of a function, stated the pigeonhole principle, and established the theory of Fourier series on rigorous ground.