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| '''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 analysis|asymptotic distribution results]]. The theory was invented and developed by [[John Knopfmacher]] in the early 1970s.
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| == Arithmetic semigroups ==
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| The fundamental notion involved is that of an '''arithmetic semigroup''', which is a [[commutative]] [[monoid]] ''G'' satisfying the following properties:
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| *There exists a [[countable]] [[subset]] (finite or countably infinite) ''P'' of ''G'', such that every element ''a'' ≠ 1 in ''G'' has a unique factorisation of the form
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| ::<math>a = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_r^{\alpha_r}</math>
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| :where the ''p''<sub>''i''</sub> are distinct elements of ''P'', the α<sub>''i''</sub> are positive [[integer]]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''.
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| *There exists a [[real number|real]]-valued ''norm mapping'' <math>|\mbox{ }|</math> on ''G'' such that
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| *#<math>|1| = 1</math>
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| *#<math>|p| > 1 \mbox{ for all } p \in P</math>
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| *#<math>|ab| = |a| |b| \mbox{ for all } a,b \in G</math>
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| *#The total number <math>N_G(x)</math> of elements <math>a \in G</math> of norm <math>|a| \leq x</math> is finite, for each real <math>x > 0</math>.
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| ===Additive number systems===
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| {{redirect|Additive number system|food additive numbering|E number}}
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| An '''additive number system''' is an arithmetic semigroup in which the underlying monoid ''G'' is [[Free commutative monoid|free abelian]]. The norm function may be written additively.<ref name=Bur20>Burris (2001) p.20</ref>
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| 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''<ref name=Bur26>Burris (2001) p.26</ref>
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| :<math>A(x) = \sum_n a(n) x^n = \prod_n (1-x^n)^{-p(n)} \ </math>
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| 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'').<ref name=Bur31>Burris (2001) p.31</ref>
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| The fundamental identity has the alternative form<ref name=Bur34>Burris (2001) p.34</ref>
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| :<math>A(x) = \exp\left({ \sum_{m \ge 1} \frac{P(x^m)}{m} }\right) \ . </math> | |
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| == Examples ==
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| *The prototypical example of an arithmetic semigroup is the multiplicative [[semigroup]] of [[negative and positive numbers|positive]] [[integer]]s ''G'' = '''Z'''<sup>+</sup> = {1, 2, 3, ...}, with subset of rational [[prime number|prime]]s ''P'' = {2, 3, 5, ...}. Here, the norm of an integer is simply <math>|n| = n</math>, so that <math>N_G(x) = \lfloor x \rfloor</math>, the [[floor function|greatest integer]] not exceeding ''x''.
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| *If ''K'' is an [[algebraic number field]], i.e. a finite extension of the [[field (mathematics)|field]] of [[rational number]]s '''Q''', then the set ''G'' of all nonzero [[ideal (ring theory)|ideal]]s in the [[ring (mathematics)|ring]] of integers ''O''<sub>''K''</sub> of ''K'' forms an arithmetic semigroup with identity element ''O''<sub>''K''</sub> and the norm of an ideal ''I'' is given by the cardinality of the quotient ring ''O''<sub>''K''</sub>/''I''. In this case, the appropriate generalisation of the prime number theorem is the ''[[Landau prime ideal theorem]]'', which describes the asymptotic distribution of the ideals in ''O''<sub>''K''</sub>.
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| *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 theory)|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.
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| **The category of all [[finite set|finite]] [[abelian group]]s under the usual direct product operation and norm mapping <math>|A| = \mbox{ card}(A)</math>. The indecomposable objects are the [[cyclic group]]s of prime power order.
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| **The category of all [[compact space|compact]] [[simply-connected]] globally symmetric Riemannian [[manifold]]s under the Riemannian product of manifolds and norm mapping <math>|M| = c^{\mbox{dim }M}</math>, 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.
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| **The category of all [[pseudometric space|pseudometrisable]] finite [[topological space]]s under the [[disjoint union (topology)|topological sum]] and norm mapping <math>|X| = 2^{\mbox{card}(X)}</math>. The indecomposable objects are the [[connected space]]s.
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| == Methods and techniques ==
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| The use of [[arithmetic function]]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:
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| *''Axiom A''. There exist positive constants ''A'' and <math>\delta</math>, and a constant <math>\nu</math> with <math>0 \le \nu < \delta</math>, such that <math>N_G(x) = Ax^{\delta} + O(x^{\nu}) \mbox { as } x \rightarrow \infin.</math><ref name=K75>Knopfmacher (1990) p.75</ref>
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| For any arithmetic semigroup which satisfies Axiom ''A'', we have the following ''abstract prime number theorem'':<ref name=K154>Knopfmacher (1990) p.154</ref>
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| :<math>\pi_G(x) \sim \frac{x^{\delta}}{\delta \log x} \mbox { as } x \rightarrow \infin</math>
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| where π<sub>''G''</sub>(''x'') = total number of elements ''p'' in ''P'' of norm |''p''| ≤ ''x''.
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| ===Arithmetical formation===
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| 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 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 group|character]] of ''A'' then we can define a [[Dirichlet series]]
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| :<math> \sum_{g \in G} \chi([g]) |g|^{-s} </math>
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| which provides a notion of zeta function for arithmetical semigroup.<ref>Knopfmacher (1990) pp.250–264</ref>
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| ==See also==
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| * [[Axiom A]], a property of dynamical systems
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| * [[Beurling zeta function]]
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Burris | first=Stanley N. | title=Number theoretic density and logical limit laws | series=Mathematical Surveys and Monographs | volume=86 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2001 | isbn=0-8218-2666-2 | zbl=0995.11001 }}
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| *{{cite book | title=Abstract Analytic Number Theory | first=John | last=Knopfmacher | edition=2nd | publisher=Dover Publishing | location=New York, NY | year=1990 | origyear=1975 | isbn=0-486-66344-2 | zbl=0743.11002 }}
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| *{{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=0-521-84903-9 | zbl=1142.11001 | page=278}}
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| [[Category:Algebraic number theory|*]]
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| [[Category:Analytic number theory|*]]
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