HOME

TheInfoList



OR:

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 Arne Carl-August Beurling (3 February 1905 – 20 November 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937–1954) and later at the Institute for Advanced Study in Princeton, New Jersey. Beurling worked ...
in the twentieth century.


Arithmetic semigroups

The fundamental notion involved is that of an arithmetic semigroup, which is a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
''G'' satisfying the following properties: *There exists a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
(finite or countably infinite) ''P'' of ''G'', such that every element ''a'' ≠ 1 in ''G'' has a unique factorisation of the form ::a = p_1^ p_2^ \cdots p_r^ :where the ''p''''i'' are distinct elements of ''P'', the α''i'' are positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
-valued ''norm mapping'' , \mbox, on ''G'' such that *#, 1, = 1 *#, p, > 1 \mbox p \in P *#, ab, = , a, , b, \mbox a,b \in G *#The total number N_G(x) of elements a \in G of norm , a, \leq x is finite, for each real x > 0.


Additive number systems

An additive number system is an arithmetic semigroup in which the underlying monoid ''G'' is free abelian. The norm function may be written additively.Burris (2001) p.20 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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
. We have the ''fundamental identity''Burris (2001) p.26 :A(x) = \sum_n a(n) x^n = \prod_n (1-x^n)^ \ 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 In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of the power series ''A''(''x'').Burris (2001) p.31 The fundamental identity has the alternative formBurris (2001) p.34 :A(x) = \exp\left(\right) \ .


Examples

*The prototypical example of an arithmetic semigroup is the multiplicative
semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...
of
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s ''G'' = Z+ = , with subset of rational
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s ''P'' = . Here, the norm of an integer is simply , n, = n, so that N_G(x) = \lfloor x \rfloor, the greatest integer not exceeding ''x''. *If ''K'' is an
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
, i.e. a finite extension of the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s Q, then the set ''G'' of all nonzero ideals in the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of integers ''OK'' of ''K'' forms an arithmetic semigroup with identity element ''OK'' and the norm of an ideal ''I'' is given by the cardinality of the quotient ring ''OK''/''I''. In this case, the appropriate generalisation of the prime number theorem is the ''
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''. Exa ...
'', which describes the asymptotic distribution of the ideals in ''OK''. *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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
, 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is com ...
s under the usual direct product operation and norm mapping , A, = \operatorname(A). The indecomposable objects are the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
s of prime power order. **The category of all
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...
globally symmetric Riemannian
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s under the Riemannian product of manifolds and norm mapping , M, = c^, 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 finite
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s under the topological sum and norm mapping , X, = 2^. The indecomposable objects are the
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 non-empty open subsets. Connectedness is one of the principal topological properties ...
s.


Methods and techniques

The use of
arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
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 \delta, and a constant \nu with 0 \le \nu < \delta, such that N_G(x) = Ax^ + O(x^) \mbox x \rightarrow \infin.Knopfmacher (1990) p.75 For any arithmetic semigroup which satisfies Axiom ''A'', we have the following ''abstract prime number theorem'':Knopfmacher (1990) p.154 :\pi_G(x) \sim \frac \mbox x \rightarrow \infin where π''G''(''x'') = 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 number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...
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 group ''A''. 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 Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of ''A'' then we can define a
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...
: \sum_ \chi( , g, ^ which provides a notion of zeta function for arithmetical semigroup.Knopfmacher (1990) pp.250–264


See also

* Axiom A, a property of dynamical systems * Beurling zeta function


References

* * *{{cite book , first1=Hugh L. , last1=Montgomery , author1-link=Hugh Montgomery (mathematician) , first2=Robert C. , last2=Vaughan , author2-link=Robert Charles Vaughan (mathematician) , title=Multiplicative number theory I. Classical theory , series=Cambridge studies in advanced mathematics , volume=97 , year=2007 , isbn=978-0-521-84903-6 , zbl=1142.11001 , page=278 * *