Why did Dirichlet's proof begin analytic number theory?

Peter Gustav Lejeune Dirichlet is credited with initiating analytic number theory due to the innovative methodology he employed to prove that any arithmetic progression $a,a+b,a+2b,\dots$ (where $a$ and $b$ are relatively prime) contains infinitely many prime numbers.12

While the theorem itself was a significant advancement in number theory, its historical importance lies in Dirichlet’s strategy: he utilized techniques from calculus and analysis to establish a fundamental result regarding the distribution of integers. By applying these continuous mathematical tools to solve a discrete problem, he introduced a powerful new approach that defined the branch of analytic number theory.13