Analytic number theory exists and involves calculus, but it's not what the linked post is about. The article talks about Hensel's lemma, which is a purely algebraic statement with a purely algebraic proof, which, however, is inspired by techniques from calculus. This is typically still categorized as algebraic number theory.
https://en.wikipedia.org/wiki/Analytic_number_theory
The prime number theorem, on how prime numbers are distributed amongst the integers, was first proved using analytic techniques.