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.
Arithmetic semigroups
The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoidG satisfying the following properties:
There exists a countablesubsetP of G, such that every element a ≠ 1 in G has a unique factorisation of the form
There exists a real-valued norm mapping on G such that
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#The total number of elements of norm is finite, for each real.
Additive number systems
An additive number system is an arithmetic semigroup in which the underlying monoidG is free abelian. The norm function may be written additively. If the norm is integer-valued, we associate counting functions a and p 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 and P 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 seriesA. The fundamental identity has the alternative form
Examples
The prototypical example of an arithmetic semigroup is the multiplicative semigroup of positive integers G = Z+ =, with subset of rational primes P =. Here, the norm of an integer is simply, so that, the greatest integer not exceeding x.
Various arithmetical categories which satisfy a theorem of Krull-Schmidt type can be considered. In all these cases, the elements of G are isomorphism classes in an appropriate category, and P consists of all isomorphism classes of indecomposable objects, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following.
*The category of all finite abelian groups under the usual direct product operation and norm mapping The indecomposable objects are the cyclic groups of prime power order.
*The category of all compactsimply-connected globally symmetric Riemannian manifolds under the Riemannian product of manifolds and norm mapping where c > 1 is fixed, and dim M denotes the manifold dimension of M. The indecomposable objects are the compact simply-connected irreducible symmetric spaces.
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:
Axiom A. There exist positive constants A and, and a constant with, such that
For any arithmetic semigroup which satisfies Axiom A, we have the following abstract prime number theorem: where πG = total number of elements p in P of norm |p| ≤ x.
Arithmetical formation
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. In the case of number fields, for example, this is Chebotarev's density theorem. An arithmetical formation is an arithmetic semigroup G with an equivalence relation ≡ such that the quotient G/≡ is a finite abelian groupA. This quotient is the class group of the formation and the equivalence classes are generalised arithmetic progressions or generalised ideal classes. If χ is a character of A then we can define a Dirichlet series which provides a notion of zeta function for arithmetical semigroup.
