The mathematical field of tackling number theory problems in this way is called analytic 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.

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.