Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. 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 fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying the following properties:

An additive number system is an arithmetic semigroup in which the underlying monoid G is free abelian. The norm function may be written additively.

If the norm is integer-valued, we associate counting functions a(n) and p(n) with G where p counts the number of elements of P of norm n, and a counts the number of elements of G of norm n. We let A(x) and P(x) be the corresponding formal power series. We have the fundamental identity

which formally encodes the unique expression of each element of G as a product of elements of P. The radius of convergence of G is the radius of convergence of the power series A(x).

The fundamental identity has the alternative form

The use of arithmetic functions and zeta functions is extensive. 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. Such a typical axiom is the following, usually called "Axiom A" in the literature:

For any arithmetic semigroup which satisfies Axiom A, we have the following abstract prime number theorem: