Part 8/9:
Dirichlet's Theorem and Its Implications
Dirichlet's theorem stands as a pivotal component in this discussion of prime distributions among residue classes. The theorem posits that there are infinitely many primes in any residue class that is co-prime to a chosen integer. The historical significance of this theorem cannot be overstated; it opens a pathway that connects residue classes with the density of prime numbers.
Alongside proofs that utilize advanced techniques from complex analysis, the understanding of how primes are distributed among residues continues to illuminate contemporary research, especially regarding prime gaps and conjectures surrounding twin primes.