How did Dirichlet prove that some arithmetic progressions contain infinitely many primes?

Peter Gustav Lejeune Dirichlet proved that any arithmetic progression $a,a+b,a+2b,\dots$ contains infinitely many prime numbers, provided that $a$ and $b$ are relatively prime (meaning they share no common factors other than 1).12

What made this proof particularly groundbreaking was Dirichlet’s methodology. Rather than relying solely on traditional number-theoretic approaches, he employed techniques from calculus to address a problem concerning integers. By applying these analytical methods, Dirichlet established a bridge between the study of functions and the distribution of primes, effectively marking the birth of analytic number theory. This strategy of using complex analysis to solve problems in number theory was later expanded upon by mathematicians such as Bernhard Riemann.2