Abstract analytic number theory is a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 which takes the ideas and techniques of classical analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

 and applies them to a variety of different mathematical fields. The classical prime number theorem

Prime number theorem

In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

 serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results

Asymptotic analysis

In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...

. The theory was invented and developed by John Knopfmacher in the early 1970s.

Arithmetic semigroups

The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

 G satisfying the following properties:

• There exists a countable subset
  Subset
  In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
  
   (finite or countably infinite) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form

where the p_i are distinct elements of P, the α_i are positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, r may depend on a, and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of P are called the primes of G.

• There exists a real
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  -valued norm mapping on G such that
  1. 
  2. 
  3. 
  4. The total number of elements of norm is finite, for each real .

Examples

• The prototypical example of an arithmetic semigroup is the multiplicative semigroup
  Semigroup
  In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
  
   of positive integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s G = Z⁺ = {1, 2, 3, ...}, with subset of rational prime
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  s P = {2, 3, 5, ...}. Here, the norm of an integer is simply , so that , the greatest integer
  Floor function
  In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
  
   not exceeding x.
• If K is an algebraic number field
  Algebraic number field
  In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
  
  , i.e. a finite extension of the field
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
   of rational number
  Rational number
  In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
  
  s Q, then the set G of all nonzero ideal
  Ideal (ring theory)
  In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
  
  s in the ring
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
   of integers O_K of K forms an arithmetic semigroup with identity element O_K and the norm of an ideal I is given by the cardinality of the quotient ring O_K/I. In this case, the appropriate generalisation of the prime number theorem is the Landau prime ideal theorem
  Landau prime ideal theorem
  In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X....
  
  , which describes the asymptotic distribution of the ideals in O_K.
• 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 group
    Abelian group
    In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
    
    s under the usual direct product operation and norm mapping . The indecomposable objects are the cyclic group
    Cyclic group
    In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
    
    s of prime power order.
  • The category of all compact
    Compact space
    In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
    
     simply-connected globally symmetric Riemannian manifold
    Manifold
    In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
    
    s 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 category of all pseudometrisable
    Pseudometric space
    In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
    
     finite topological space
    Topological space
    Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
    
    s under the topological sum
    Disjoint union (topology)
    In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
    
     and norm mapping . The indecomposable objects are the connected space
    Connected space
    In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
    
    s.

Methods and techniques

The use of arithmetic function

Arithmetic function

In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...

s 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 π_G(x) = total number of elements p in P of norm |p| ≤ x.

The notion of arithmetical formation provides a generalisation of the ideal class group

Ideal class group

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...

 in algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

 and allows for abstract asymptotic distribution results under constraints. In the case of number fields, for example, this is Chebotarev's density theorem

Chebotarev's density theorem

Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only...

.