The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results.
The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century.
The idea is to extend the various arguments and techniques of arithmetic functions and zeta functions in classical analytic number theory to the context of an arbitrary arithmetic semigroup which may satisfy one or more additional axioms.
The notion of arithmetical formation provides a generalisation of the ideal class group in algebraic number theory and allows for abstract asymptotic distribution results under constraints.
If χ is a character of A then we can define a Dirichlet series which provides a notion of zeta function for arithmetical semigroup.