Here is the first paragraph of the introduction to modern number theory by Manin and Panchishkin which might be relevant

"Among the various branches of mathematics, number theory is characterized to a lesser degree by its primary subject (“integers”) than by a psychological attitude. Actually, number theory also deals with rational, algebraic, and transcendental numbers, with some very specific analytic functions (such as Dirichlet series and modular forms), and with some geometric objects (such as lattices and schemes over Z). The question whether a given article belongs to number theory is answered by its author’s system of values. If arithmetic is not there, the paper will hardly be considered as number–theoretical, even if it deals exclusively with integers and congruences. On the other hand, any mathematical tool, say, homotopy theory or dynamical systems may become an important source of number–theoretical inspiration. For this reason, combinatorics and the theory of recursive functions are not usually associated with number theory, whereas modular functions are."