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The central concept in abstract analytic number theory is that of an arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300501.png" /> (defined below). It turns out that the study of such semi-groups and of (real- or complex-valued) functions on them makes it possible on the one hand to apply methods of classical [[Analytic number theory|analytic number theory]] in a unified way to a variety of asymptotic enumeration questions for isomorphism classes of different kinds of explicit mathematical objects. On the other hand, these procedures also lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration questions about the (mostly non-arithmetical) concrete types of mathematical objects just alluded to.
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==Arithmetical semi-groups.==
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An arithmetical semi-group is, by definition, a commutative [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300502.png" /> with identity element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300503.png" />, which contains a countable subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300504.png" /> such that every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300505.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300506.png" /> admits a unique factorization into a finite product of powers of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300507.png" />, together with a real-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300508.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a1300509.png" /> such that:
 
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005012.png" />;
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{{TEX|semi-auto}}{{TEX|part}}
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The central concept in abstract analytic number theory is that of an arithmetical semi-group $G$ (defined below). It turns out that the study of such semi-groups and of (real- or complex-valued) functions on them makes it possible on the one hand to apply methods of classical [[Analytic number theory|analytic number theory]] in a unified way to a variety of asymptotic enumeration questions for isomorphism classes of different kinds of explicit mathematical objects. On the other hand, these procedures also lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration questions about the (mostly non-arithmetical) concrete types of mathematical objects just alluded to.
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005014.png" />;
 
  
iii) the total number of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005016.png" /> is finite, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005017.png" />.
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==Arithmetical semi-groups==
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An arithmetical semi-group is, by definition, a commutative [[Semi-group|semi-group]] $G$ with identity element $1$, which contains a countable subset $P$ such that every element $a \ne 1$ in $G$ admits a unique factorization into a finite product of powers of elements of $P$, together with a real-valued mapping $\vert \cdot \vert$ on $G$ such that:
  
The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005018.png" /> are called the primes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005020.png" /> is called the norm mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005021.png" />. It is obvious that, corresponding to any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005022.png" />, the definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005023.png" /> yields a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005025.png" /> such that:
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i) $|1| = 1$, $|p| > 1$ for $p \in P$;
  
A) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005028.png" />;
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ii) $|a b| = |a| \cdot |b|$ for all $a\,,b \in G$;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005030.png" />;
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iii) the total number of elements $a$ with $|a| < x$ is finite, for each $x > 0$.
  
C) the total number of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005032.png" /> is finite, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005033.png" />.
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The elements of $P$ are called the primes of $G$, and $\vert \cdot \vert$ is called the norm mapping on $G$. It is obvious that, corresponding to any fixed $c>1$, the definition $\partial(a) = \log_c|a|$ yields a mapping $\partial$ on $G$ such that:
  
Conversely, any real-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005034.png" /> with the properties A)–C) yields a norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005035.png" />, if one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005036.png" />. In cases where such a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005037.png" /> is of primary interest, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005038.png" /> together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005039.png" /> is called an additive arithmetical semi-group, and one refers to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005040.png" /> as the degree mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005041.png" />. In most concrete examples of interest, it turns out that the norm or degree mappings represent natural  "size"  or  "dimension"  measures which are integer-valued. With an eye to applications to natural examples there is therefore little loss in henceforth restricting attention to either a single integer-valued norm mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005042.png" />, or a single integer-valued degree mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005043.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005044.png" />. Depending on which case is being considered, special interest then attaches to the basic counting functions (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005045.png" />)
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A) $\partial(1)=0$, $\partial(p)>0$ for $p \in P$;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005046.png" /></td> </tr></table>
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B) $\partial(ab) = \partial(a) + \partial(b)$ for all $a\,,b \in G$;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005047.png" /></td> </tr></table>
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C) the total number of elements $a$ with $\partial(a) \le x$ is finite, for each $x > 0$.
  
(or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005049.png" />, in the additive case).
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Conversely, any real-valued mapping $\partial$ with the properties A)–C) yields a norm on $G$, if one defines $|a| = c^{\partial(a)}$. In cases where such a mapping $\partial$ is of primary interest, $G$ together with $\partial$ is called an additive arithmetical semi-group, and one refers to $\partial$ as the degree mapping on $G$. In most concrete examples of interest, it turns out that the norm or degree mappings represent natural  "size" or  "dimension" measures which are integer-valued. With an eye to applications to natural examples there is therefore little loss in henceforth restricting attention to either a single integer-valued norm mapping $\vert\cdot\vert$, or a single integer-valued degree mapping $\partial$, on $G$. Depending on which case is being considered, special interest then attaches to the basic counting functions (for $n \in \mathbb{Z}$)
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$$
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G(n) = \#\{ a \in G : |a| = n \}
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$$
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$$
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P(n) = \#\{ p \in P : |p| = n \}
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$$
  
The prototype of all arithmetical semi-groups is of course the multiplicative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005050.png" /> of all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005051.png" />, with its subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005052.png" /> of all rational prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005053.png" />. Here one may define the norm of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005054.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005055.png" />, so that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005057.png" />.
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(or $G^\sharp(n) = \#\{ a \in G : \partial(a) = n \}$, $P^\sharp(n) = \#\{ p \in P : \partial(p) = n \}$, in the additive case).
  
The asymptotic behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005058.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005059.png" /> forms the content of the famous prime number theorem, which states that
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The prototype of all arithmetical semi-groups is of course the multiplicative semi-group $\mathbb{N}$ of all positive integers $\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime numbers $\{2,3,5,7,\ldots\}$. Here one may define the norm of an integer $|n|$ to be $n$, so that the number $\mathbb{N}(n) = 1$ for $n \ge 1$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005060.png" /></td> </tr></table>
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The asymptotic behaviour of $\pi(X) = \sum_{n \le x} P_{\mathbb{N}}(n)$ for large $x$ forms the content of the famous [[prime number theorem]], which states that
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$$
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\pi(x) \sim \frac{x}{\log x} \ \text{as}\  x \rightarrow \infty
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$$
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(cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]). A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semi-groups. For example, it is true for the multiplicative semi-group $G_K$ of all non-zero ideals in the [[Ring|ring]] $R = R(K)$ of all algebraic integers in a given [[Algebraic number|algebraic number]] field $K$, with $|I| = \mathop{card}(R/I)$ for any non-zero ideal $I$ in $R$. Here, the prime ideals act as prime elements of the semi-group $G_K$.
  
(cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]). A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semi-groups. For example, it is true for the multiplicative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005061.png" /> of all non-zero ideals in the [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005062.png" /> of all algebraic integers in a given [[Algebraic number|algebraic number]] field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005063.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005064.png" /> for any non-zero ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005065.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005066.png" />. Here, the prime ideals act as prime elements of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005067.png" />.
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A simple but nevertheless interesting example of an additive arithmetical semi-group is provided by the multiplicative semi-group $G-q$ of all monic polynomials in one indeterminate $X$ over a [[Finite field|finite field]] $\mathbb{F}_q$ with $q$ elements, with $\partial a = \mathrm{deg}\, a$ and the set $P_q$ of prime elements represented by the irreducible polynomials (cf. also [[Irreducible polynomial|Irreducible polynomial]]). Here, $G_q^\sharp(n) = q^n$, and it can be proved that
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$$
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P_q^\sharp (n) = \frac{1}{n} \sum_{r | n} \mu(r) q^{n/r}
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$$
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where $\mu$ is the classical [[Möbius function]] on $\mathbb{N}$.
  
A simple but nevertheless interesting example of an additive arithmetical semi-group is provided by the multiplicative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005068.png" /> of all monic polynomials in one indeterminate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005069.png" /> over a [[Finite field|finite field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005070.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005071.png" /> elements, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005072.png" /> and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005073.png" /> of prime elements represented by the irreducible polynomials (cf. also [[Irreducible polynomial|Irreducible polynomial]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005074.png" />, and it can be proved that
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Up to isomorphism, $G_q$ is the simplest special case of the semi-group $G_R$ of all non-zero ideals in the ring $R = R(K)$ of all integral functions in an algebraic function field $K$ in one variable $K$ over $\mathbb{F}_q$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005075.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005076.png" /> is the classical [[Möbius function|Möbius function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005077.png" />.
 
 
 
Up to isomorphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005078.png" /> is the simplest special case of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005079.png" /> of all non-zero ideals in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005080.png" /> of all integral functions in an algebraic function field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005081.png" /> in one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005082.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005083.png" />.
 
  
 
===Arithmetical categories of semi-groups.===
 
===Arithmetical categories of semi-groups.===
 
Many interesting examples of concrete, but non-classical, arithmetical semi-groups can be found by considering certain specific classes of mathematical objects, such as groups, rings, topological spaces, and so on, together with appropriate  "direct product"  operations and isomorphism relations for those classes. It is convenient, though admittedly not quite precise, to temporarily ignore the corresponding morphisms and refer to such classes of objects as  "categories"  (cf. also [[Category|Category]]).
 
Many interesting examples of concrete, but non-classical, arithmetical semi-groups can be found by considering certain specific classes of mathematical objects, such as groups, rings, topological spaces, and so on, together with appropriate  "direct product"  operations and isomorphism relations for those classes. It is convenient, though admittedly not quite precise, to temporarily ignore the corresponding morphisms and refer to such classes of objects as  "categories"  (cf. also [[Category|Category]]).
  
Now consider some category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005084.png" /> which admits a direct  "product"  (or  "sum" ) operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005085.png" /> on its objects. Suppose that this operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005086.png" /> preserves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005087.png" />-isomorphisms, is commutative and associative up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005088.png" />-isomorphism, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005089.png" /> contains a  "zero"  object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005090.png" /> (unique up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005091.png" />-isomorphism) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005092.png" /> for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005093.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005094.png" />. Then suppose that a theorem of Krull–Schmidt type is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005095.png" />, i.e., suppose that every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005096.png" /> can be expressed as a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005097.png" />-product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005098.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a13005099.png" /> that are indecomposable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050100.png" />, in a way that is unique up to permutation of terms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050101.png" />-isomorphism. In most natural situations at least, one may reformulate these conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050102.png" /> by stating that the various isomorphism classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050103.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050105.png" /> form a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050106.png" /> that is
+
Now consider some category $\mathcal{C}$ which admits a direct  "product"  (or  "sum" ) operation $\times$ on its objects. Suppose that this operation $\times$ preserves $\mathcal{C}$-isomorphisms, is commutative and associative up to $\mathcal{C}$-isomorphism, and that $\mathcal{C}$ contains a  "zero"  object $0$ (unique up to $\mathcal{C}$-isomorphism) such that $A \times 0 \cong A$ for all objects $A$ in $\mathcal{C}$. Then suppose that a theorem of Krull–Schmidt type is valid for $\mathcal{C}$, i.e., suppose that every object $A$ can be expressed as a finite $\times$-product $A \cong P_1 \times \cdots \times P_k$ of objects $P_i$ that are indecomposable with respect to $\times$, in a way that is unique up to permutation of terms and $\mathcal{C}$-isomorphism. In most natural situations at least, one may reformulate these conditions on $\mathcal{C}$ by stating that the various isomorphism classes $\bar A$ of objects $A$ in $\mathcal{C}$ form a set $G_{\mathcal{C}}$ that is
  
i) a commutative semi-group with identity with respect to the multiplication operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050107.png" />;
+
i) a commutative semi-group with identity with respect to the multiplication operation $\bar A \times \bar B = \overline{A \times B}$;
  
ii) a semi-group with the unique factorization property with respect to the isomorphism classes of the indecomposable objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050108.png" />.
+
ii) a semi-group with the unique factorization property with respect to the isomorphism classes of the indecomposable objects in $\mathcal{C}$.
  
For this reason, one may call the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050109.png" />-isomorphism classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050110.png" /> of indecomposable objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050111.png" /> the  "primes"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050112.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050113.png" />.
+
For this reason, one may call the $\mathcal{C}$-isomorphism classes $\bar P_i$ of indecomposable objects $P$ the  "primes"  of $\mathcal{C}$ or $G_{\mathcal{C}}$.
  
In many interesting cases (some of which are illustrated below), the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050114.png" /> also admits a  "norm"  function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050115.png" /> on objects which is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050116.png" />-isomorphism and has the following properties:
+
In many interesting cases (some of which are illustrated below), the category $\mathcal{C}$ also admits a  "norm"  function $\vert \cdot \vert$ on objects which is invariant under $\mathcal{C}$-isomorphism and has the following properties:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050118.png" /> for every indecomposable object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050119.png" />;
+
i) $|0| = 1$, $|P| > 1$ for every indecomposable object $P$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050120.png" /> for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050122.png" />;
+
ii) $|A \times B| = |A|.|B|$ for all objects $A$, $B$;
  
iii) the total number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050123.png" />-isomorphism classes of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050124.png" /> of norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050125.png" /> is finite, for each real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050126.png" />.
+
iii) the total number of $\mathcal{C}$-isomorphism classes of objects $A$ of norm $|A| \le x$ is finite, for each real $x > 0$.
  
Obviously, in such circumstances, the definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050127.png" /> provides a norm function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050128.png" /> satisfying the required conditions for an arithmetical semi-group. For these reasons, a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050129.png" /> with such further properties may be called an arithmetical category.
+
Obviously, in such circumstances, the definition $|\bar A| = |A|$ provides a norm function on $G_{\mathcal{C}}$ satisfying the required conditions for an arithmetical semi-group. For these reasons, a category $\mathcal{C}$ with such further properties may be called an arithmetical category.
  
 
Now consider some concrete illustrations for the above concepts, taken from [[#References|[a2]]], [[#References|[a3]]].
 
Now consider some concrete illustrations for the above concepts, taken from [[#References|[a2]]], [[#References|[a3]]].
  
a) (Finite Abelian groups; cf. [[Abelian group|Abelian group]].) One of the simplest non-trivial examples of an arithmetical category is provided by the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050130.png" /> of all finite Abelian groups, together with the usual direct product operation and the norm function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050131.png" />. Here, the Krull–Schmidt theorem reduces to the well-known fundamental theorem on finite Abelian groups, the indecomposable objects of this kind being simply the various cyclic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050132.png" /> of prime-power order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050133.png" /> (cf. also [[Cyclic group|Cyclic group]]).
+
a) (Finite Abelian groups; cf. [[Abelian group]].) One of the simplest non-trivial examples of an arithmetical category is provided by the category $\mathcal{A}$ of all finite Abelian groups, together with the usual direct product operation and the norm function $|A| = \mathrm{card}(A)$. Here, the Krull–Schmidt theorem reduces to the well-known fundamental theorem on finite Abelian groups, the indecomposable objects of this kind being simply the various [[cyclic group]]s $C_{p^n}$ of prime-power order $p^n$.
  
b) The category of all semi-simple associative rings of finite cardinality (cf. also [[Associative rings and algebras|Associative rings and algebras]]).
+
b) The category of all semi-simple associative rings of finite cardinality (cf. also [[Associative rings and algebras]]).
  
c) The category of all semi-simple finite-dimensional associate algebras over a given field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050134.png" /> (cf. also [[Associative rings and algebras|Associative rings and algebras]]; [[Semi-simple ring|Semi-simple ring]]).
+
c) The category of all semi-simple finite-dimensional associate algebras over a given field $F$ (cf. also [[Associative rings and algebras]]; [[Semi-simple ring]]).
  
d) The category of all semi-simple finite-dimensional Lie algebras over a given field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050135.png" /> (cf. also [[Lie algebra|Lie algebra]]).
+
d) The category of all semi-simple finite-dimensional [[Lie algebra]]s over a given field $F$.
  
e) The category of all compact simply-connected globally symmetric Riemannian manifolds (cf. also [[Globally symmetric Riemannian space|Globally symmetric Riemannian space]]).
+
e) The category of all compact simply-connected globally symmetric Riemannian manifolds (cf. also [[Globally symmetric Riemannian space]]).
  
f) The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050136.png" /> of topological spaces of finite cardinality (cf. also [[Topological space|Topological space]]) with the property that a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050137.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050138.png" /> if and only if each connected component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050139.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050140.png" />.
+
f) The category $\mathcal{T}$ of [[topological space]]s of finite cardinality with the property that a space $Y$ lies in $\mathcal{T}$ if and only if each connected component of $Y$ lies in $\mathcal{T}$.
  
 
==Zeta-functions and enumeration problems.==
 
==Zeta-functions and enumeration problems.==
For a given arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050141.png" />, information on the basic counting functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050142.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050143.png" /> can often be obtained, algebraically or with the aid of analysis, via a certain series-production relation called the Euler product formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050144.png" />.
+
For a given arithmetical semi-group $G$, information on the basic counting functions $G ( n )$, $P ( n )$ can often be obtained, algebraically or with the aid of analysis, via a certain series-production relation called the Euler product formula for $G$.
  
Indeed, ignoring questions of convergence for the moment, note that (by the unique factorization into prime elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050145.png" />) the series
+
Indeed, ignoring questions of convergence for the moment, note that (by the unique factorization into prime elements of $G$) the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050146.png" /></td> </tr></table>
+
\begin{equation*} \zeta _ { G } ( z ) = \sum _ { n = 1 } ^ { \infty } G ( n ) n ^ { - z } = \sum _ { a \in G } | a | ^ { - z } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050147.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050147.png"/></td> </tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050148.png" /></td> </tr></table>
+
\begin{equation*} = 1 + \sum | p _ { 1 } | ^ { - r _ { 1 } z } \ldots | p _ {m } | ^ { - r _ { m } z } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050149.png" /></td> </tr></table>
+
\begin{equation*} = \prod _ { p \in P } ( 1 + | p | ^ { - z } + | p | ^ { - 2 z } + \ldots ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050150.png" /></td> </tr></table>
+
\begin{equation*} = \prod _ { p \in P } ( 1 - | p | ^ { - z } ) ^ { - 1 } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050151.png" /></td> </tr></table>
+
\begin{equation*} = \prod _ { m = 2 } ^ { \infty } ( 1 - m ^ { - z } ) ^ { - P ( m ) }. \end{equation*}
  
As a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050153.png" /> is called the zeta-function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050154.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050155.png" /> is an additive arithmetical semi-group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050156.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050157.png" />, one may substitute the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050158.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050159.png" /> and obtain the modified Euler product formula:
+
As a function of $z$, $\zeta _ { G } ( z )$ is called the zeta-function of $G$. If $G$ is an additive arithmetical semi-group with $| a | = c ^ { \partial ( a ) }$ for some integer $c > 1$, one may substitute the symbol $y$ for $c ^ { - z }$ and obtain the modified Euler product formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050160.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { n = 0 } ^ { \infty } G ^ { \# } ( n ) y ^ { n } = \prod _ { m = 1 } ^ { \infty } ( 1 - y ^ { m } ) ^ { - P ^ { \# } ( m ) }; \end{equation*}
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050161.png" /> is called the modified zeta-function (or generating function) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050162.png" />.
+
then $Z _ { G } ( y ) = \sum _ { n = 0 } ^ { \infty } G ^ { \# } ( n ) y ^ { n }$ is called the modified zeta-function (or generating function) of $G$.
  
 
Some explicit illustrations of zeta-functions and Euler products are given below.
 
Some explicit illustrations of zeta-functions and Euler products are given below.
  
 
===The Riemann zeta-function.===
 
===The Riemann zeta-function.===
For the basic semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050163.png" /> of positive integers, the [[Zeta-function|zeta-function]] is
+
For the basic semi-group $\mathbf{N}$ of positive integers, the [[zeta-function]] is
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050164.png" /></td> </tr></table>
+
\zeta(z) = \sum_{n=1}^\infty n^{-z} \ ;
 
+
$$
it is called the Riemann zeta-function, and the classical Euler product formula reads:
+
it is called the Riemann zeta-function, and the classical [[Euler product]] formula reads:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050165.png" /></td> </tr></table>
+
\zeta(z) = \prod_{\text{primes}\,p\,\text{of}\,\mathbf{N}} \left({ 1 - p^{-z} }\right)^{-1} \ .
 +
$$
  
 
===The Dedekind zeta-function.===
 
===The Dedekind zeta-function.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050166.png" /> denote the (above-mentioned) arithmetical semi-group of all non-zero  "integral"  ideals in a given algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050167.png" />. The zeta-function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050168.png" /> is then
+
Let $G _ { K }$ denote the (above-mentioned) arithmetical semi-group of all non-zero  "integral"  ideals in a given algebraic number field $K$. The zeta-function for $G _ { K }$ is then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050169.png" /></td> </tr></table>
+
\begin{equation*} \zeta _ { K } ( z ) = \sum _ { I \in G _ { K } } | I | ^ { - z } = \sum _ { n = 1 } ^ { \infty } K ( n ) n ^ { - z }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050170.png" /> denotes the total number of ideals of norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050171.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050172.png" />; it is known as the Dedekind zeta-function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050173.png" />. (See also [[Zeta-function|Zeta-function]].)
+
where $K ( n )$ denotes the total number of ideals of norm $n$ in $G _ { K }$; it is known as the Dedekind zeta-function of $K$. (See also [[Zeta-function|Zeta-function]].)
  
 
===Monic polynomials over a finite field.===
 
===Monic polynomials over a finite field.===
For the additive arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050174.png" /> of all monic polynomials in one indeterminate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050175.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050176.png" /> (see above), the generating function may be written as
+
For the additive arithmetical semi-group $G_q$ of all monic polynomials in one indeterminate $X$ over $\mathbf{F} _ { q }$ (see above), the generating function may be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050177.png" /></td> </tr></table>
+
\begin{equation*} Z _ { q } ( y ) = \sum _ { n = 0 } ^ { \infty } q ^ { n } y ^ { n } = ( 1 - q y ) ^ { - 1 }, \end{equation*}
  
and the above-mentioned explicit formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050178.png" /> can be deduced as an algebraic consequence of the Euler product for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050179.png" />.
+
and the above-mentioned explicit formula for $P _ { q } ^ { \# } ( n )$ can be deduced as an algebraic consequence of the Euler product for $G_q$.
  
 
===Finite Abelian groups.===
 
===Finite Abelian groups.===
For the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050180.png" /> of all finite Abelian groups, the zeta-function may be written as
+
For the category $\mathcal{A}$ of all finite Abelian groups, the zeta-function may be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050181.png" /></td> </tr></table>
+
\begin{equation*} \zeta _ { A } ( z ) = \sum _ { n = 1 } ^ { \infty } a ( n ) n ^ { - z }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050182.png" /> denotes the total number of isomorphism classes of Abelian groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050183.png" />. The discussion of  "primes"  in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050184.png" /> given above shows that here the Euler product may be written as a double product
+
where $a ( n )$ denotes the total number of isomorphism classes of Abelian groups of order $n$. The discussion of  "primes"  in $\mathcal{A}$ given above shows that here the Euler product may be written as a double product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050185.png" /></td> </tr></table>
+
\begin{equation*} \zeta _ { A } ( z ) = \prod _ {\substack{ r \geq 1 \\ \text{primes } p \in \mathbf{N}}} \quad ( 1 - p ^ { - r z } ) ^ { - 1 } = \prod _ { r = 1 } ^ { \infty } \zeta ( r z ), \end{equation*}
  
 
by the Euler product formula for the Riemann zeta-function.
 
by the Euler product formula for the Riemann zeta-function.
  
For the subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050186.png" /> of all finite Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050187.png" />-groups, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050188.png" /> is a fixed prime number (cf. also [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050189.png" />-group]]), it is natural to regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050190.png" /> as an additive arithmetical category, with degree mapping defined by
+
For the subcategory $\mathcal{A} ( p )$ of all finite Abelian $p$-groups, where $p$ is a fixed prime number (cf. also [[P-group|$p$-group]]), it is natural to regard $\mathcal{A} ( p )$ as an additive arithmetical category, with degree mapping defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050191.png" /></td> </tr></table>
+
\begin{equation*} \partial ( A ) = \operatorname { log } _ { p } \operatorname { card } ( A ). \end{equation*}
  
In that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050192.png" /> has exactly one prime of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050193.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050194.png" />. Therefore the Euler product formula implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050195.png" /> has the generating function
+
In that case, $\mathcal{A} ( p )$ has exactly one prime of degree $r$ for each $r = 1,2 , \dots$. Therefore the Euler product formula implies that $\mathcal{A} ( p )$ has the generating function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050196.png" /></td> </tr></table>
+
\begin{equation*} Z _ { \mathcal{A} ( p ) } ( y ) = \prod _ { r = 1 } ^ { \infty } ( 1 - y ^ { r } ) ^ { - 1 } = \sum _ { n = 0 } ^ { \infty } \mathbf{p} ( n ) y ^ { n }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050197.png" /> is the total number of isomorphism classes of Abelian groups of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050198.png" /> in the above sense. In fact, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050200.png" /> equals the total number of ways of partitioning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050201.png" /> into a sum of positive integers, which is also the number of pseudo-metrizable finite topological spaces of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050202.png" /> (see f) above). Thus, the corresponding latter category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050203.png" /> (say) has the same generating function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050204.png" />.
+
where $\mathbf{ p} ( n ) = a ( p ^ { n } )$ is the total number of isomorphism classes of Abelian groups of degree $n$ in the above sense. In fact, for $n > 0$, $\mathbf{p} ( n )$ equals the total number of ways of partitioning $n$ into a sum of positive integers, which is also the number of pseudo-metrizable finite topological spaces of cardinality $n$ (see f) above). Thus, the corresponding latter category $\mathcal{P}$ (say) has the same generating function as $\mathcal{A} ( p )$.
  
 
==Types of arithmetical semi-groups.==
 
==Types of arithmetical semi-groups.==
 
Bearing in mind the emphasis on concrete realizations of arithmetical semi-groups in a variety of areas of mathematics, it is reasonable to classify them and to base further investigations according to common features which may be exhibited by the initial enumeration theorems for particular sets of examples. In that way, further questions and enumeration problems may be investigated uniformly under suitable covering assumptions or  "axioms"  appropriate for particular natural sets of examples. On this basis, a small number of special types of arithmetical semi-groups have so far (2000) been found to predominate amongst natural concrete examples.
 
Bearing in mind the emphasis on concrete realizations of arithmetical semi-groups in a variety of areas of mathematics, it is reasonable to classify them and to base further investigations according to common features which may be exhibited by the initial enumeration theorems for particular sets of examples. In that way, further questions and enumeration problems may be investigated uniformly under suitable covering assumptions or  "axioms"  appropriate for particular natural sets of examples. On this basis, a small number of special types of arithmetical semi-groups have so far (2000) been found to predominate amongst natural concrete examples.
  
===Classical and axiom-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050205.png" /> type semi-groups.===
+
===Classical and axiom-$A$ type semi-groups.===
 
The strictly classical arithmetical semi-groups of analytic number theory are the multiplicative semi-group of all positive integers and the multiplicative semi-group of all non-zero ideals in the ring of all algebraic integers in a given algebraic number field (see above). For example, H. Weber and E. Landau proved theorems to the effect that
 
The strictly classical arithmetical semi-groups of analytic number theory are the multiplicative semi-group of all positive integers and the multiplicative semi-group of all non-zero ideals in the ring of all algebraic integers in a given algebraic number field (see above). For example, H. Weber and E. Landau proved theorems to the effect that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \sum _ { n \leq x } G _ { K } ( n ) = A _ { K } x + O ( x ^ { \eta_K} ) \text { as } x \rightarrow \infty, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050207.png" /> is the semi-group of all  "integral"  ideals in a given algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050208.png" />. Landau in particular used (a1) in order to extend many asymptotic results about arithmetical functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050209.png" /> to similar functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050210.png" />.
+
where $G _ { K }$ is the semi-group of all  "integral"  ideals in a given algebraic number field $K$. Landau in particular used (a1) in order to extend many asymptotic results about arithmetical functions on $\mathbf{N}$ to similar functions on $G _ { K }$.
  
In quite a different direction, P. Erdős and G. Szekeres proved in 1934 for the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050211.png" /> of all finite Abelian groups that
+
In quite a different direction, P. Erdős and G. Szekeres proved in 1934 for the category $\mathcal{A}$ of all finite Abelian groups that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050212.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \sum _ { n \leq x } a ( n ) = A _ { 1 } x + O ( \sqrt { x } ) \quad \text { as } x \rightarrow \infty, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050213.png" />.
+
where $A _ { 1 } = \prod _ { r \leq 2 } \zeta ( r ) = 2.29\dots$.
  
At a later stage, for the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050214.png" /> of semi-simple finite rings, I.G. Connell and J. Knopfmacher independently proved that
+
At a later stage, for the category $\mathcal{S}$ of semi-simple finite rings, I.G. Connell and J. Knopfmacher independently proved that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050215.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \sum _ { n \leq x } S ( n ) = A _ { 2 } x + O ( \sqrt { x } ) \quad \text { as } x \rightarrow \infty, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050216.png" />.
+
where $A _ { 2 } = \prod _ { rm ^ { 2 } \geq 2 } ^ { 2 }  \zeta ( rm ^ { 2 } ) = 2.49 \dots$.
  
Strong concrete motivation was available for unifying certain further developments under the umbrella of general studies of an abstract arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050217.png" /> satisfying the so-called axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050219.png" />: There exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050220.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050221.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050222.png" /> (all depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050223.png" />), such that
+
Strong concrete motivation was available for unifying certain further developments under the umbrella of general studies of an abstract arithmetical semi-group $G$ satisfying the so-called axiom $A$: There exist constants $A _ { G } > 0$, $\delta > 0$ and $\eta < \delta$ (all depending on $G$), such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050224.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { n \leq x } G ( n ) = A _ { G } x ^ { \delta } + O ( x ^ { \eta } ) \text { as } x \rightarrow \infty . \end{equation*}
  
Theorems based on the assumption of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050225.png" /> often simultaneously generalize earlier results for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050227.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050228.png" />, and provide additional asymptotic enumeration theorems for a variety of arithmetical categories like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050229.png" /> and many others.
+
Theorems based on the assumption of axiom $A$ often simultaneously generalize earlier results for $\mathbf{N}$, $G _ { K }$ and $G_{\mathcal{A}}$, and provide additional asymptotic enumeration theorems for a variety of arithmetical categories like $\mathcal{S}$ and many others.
  
===Axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050230.png" /> type semi-groups.===
+
===Axiom $A ^ { \# }$ type semi-groups.===
Consideration of the examples of multiplicative semi-groups of monic polynomials in one indeterminate, and also of enumeration theorems for some infinite families of explicit additive arithmetical categories connected with rings of integral functions in algebraic function fields over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050231.png" /> (cf. [[#References|[a4]]], [[#References|[a5]]]), provides a wealth of motivation for studying an abstract additive arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050232.png" /> satisfying axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050234.png" />: There exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050235.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050236.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050237.png" /> (all depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050238.png" />) such that
+
Consideration of the examples of multiplicative semi-groups of monic polynomials in one indeterminate, and also of enumeration theorems for some infinite families of explicit additive arithmetical categories connected with rings of integral functions in algebraic function fields over $\mathbf{F} _ { q }$ (cf. [[#References|[a4]]], [[#References|[a5]]]), provides a wealth of motivation for studying an abstract additive arithmetical semi-group $G$ satisfying axiom $A ^ { \# }$: There exist constants $A _ { G } > 0$, $q > 1$ and $\nu < 1$ (all depending on $G$) such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050239.png" /></td> </tr></table>
+
\begin{equation*} G ^ { \# } ( n ) = A _ { G } q ^ { n } + O ( q ^ { \nu  n } ) \text { as } n \rightarrow \infty. \end{equation*}
  
With this axiom as a basis instead of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050240.png" />, problems similar to those outlined above may be investigated, with similar motivation to those stimulating the axiom-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050241.png" /> type studies. It then turns out that the ensuing results and methods of proof sometimes but not always possess parallels to those subject to axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050242.png" />.
+
With this axiom as a basis instead of axiom $A$, problems similar to those outlined above may be investigated, with similar motivation to those stimulating the axiom-$A$ type studies. It then turns out that the ensuing results and methods of proof sometimes but not always possess parallels to those subject to axiom $A$.
  
A curious illustration of a non-parallel result arises with the abstract prime number theorem (or abstract prime element theorem) subject to axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050243.png" />. In 1976, Knopfmacher derived such a theorem, on the initial foundation of some plausible-looking lemmas parallel to ones under axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050244.png" />. However, in 1989 and later, other authors independently found and then closed certain gaps in those lemmas. The combined efforts of various authors then led to a final theorem with two cases, depending on whether or not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050245.png" />; contributions to this were made by S.D. Cohen, K.-H. Indlekofer, E. Manstavičius, R. Warlimont and W.-B. Zhang (see e.g. [[#References|[a1]]], [[#References|[a4]]]).
+
A curious illustration of a non-parallel result arises with the abstract prime number theorem (or abstract prime element theorem) subject to axiom $A ^ { \# }$. In 1976, Knopfmacher derived such a theorem, on the initial foundation of some plausible-looking lemmas parallel to ones under axiom $A$. However, in 1989 and later, other authors independently found and then closed certain gaps in those lemmas. The combined efforts of various authors then led to a final theorem with two cases, depending on whether or not $Z _ { G } ( - q ^ { - 1 } ) = 0$; contributions to this were made by S.D. Cohen, K.-H. Indlekofer, E. Manstavičius, R. Warlimont and W.-B. Zhang (see e.g. [[#References|[a1]]], [[#References|[a4]]]).
  
A strange point about this result is that the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050246.png" /> holds for all the natural examples which initially motivated axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050247.png" />. Although ingenious examples in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050248.png" /> have also been constructed, those found up to now might be viewed as somewhat pathological or contrived. Therefore, in terms of the  "natural-example-based approach"  to this subject outlined in the beginning, it would not be unreasonable to continue the present (2000) direction of investigation under the combined assumption of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050249.png" /> with the additional axiom
+
A strange point about this result is that the case $Z _ { G } ( - q ^ { - 1 } ) \neq 0$ holds for all the natural examples which initially motivated axiom $A ^ { \# }$. Although ingenious examples in which $Z _ { G } ( - q ^ { - 1 } ) = 0$ have also been constructed, those found up to now might be viewed as somewhat pathological or contrived. Therefore, in terms of the  "natural-example-based approach"  to this subject outlined in the beginning, it would not be unreasonable to continue the present (2000) direction of investigation under the combined assumption of axiom $A ^ { \# }$ with the additional axiom
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050250.png" /></td> </tr></table>
+
\begin{equation*} Z _ { G } ( - q ^ { - 1 } ) \neq 0. \end{equation*}
  
In fact (see e.g. [[#References|[a3]]], [[#References|[a5]]]) many consequences of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050251.png" /> are unrelated to the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050252.png" />, and so the simplifying additional axiom would only sometimes become relevant (but nevertheless reasonable to then assume at such a stage).
+
In fact (see e.g. [[#References|[a3]]], [[#References|[a5]]]) many consequences of axiom $A ^ { \# }$ are unrelated to the value of $Z _ { G } ( - q ^ { - 1 } )$, and so the simplifying additional axiom would only sometimes become relevant (but nevertheless reasonable to then assume at such a stage).
  
===Axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050253.png" />.===
+
===Axiom $C$.===
The examples listed earlier included many involving an additive arithmetical category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050254.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050255.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050256.png" /> have quite a different behaviour from that given by axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050257.png" />. Here, although the objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050258.png" /> may sometimes be rather complicated, the presently (as of 2000) known structure theorems for those objects often lead to a relatively simple estimation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050259.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050260.png" />. Surprisingly perhaps, it turns out that sharp asymptotic information can then be deduced about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050261.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050262.png" /> by methods of classical-type arithmetical partition theory, which were initiated by G.H. Hardy and G. Ramanujan in 1917. These methods belong to a quite different branch of classical [[Analytic number theory|analytic number theory]] from those involved in the earlier discussion of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050263.png" />.
+
The examples listed earlier included many involving an additive arithmetical category $\mathcal{C}$ for which $G _ { \cal C } ^ { \# } ( n )$ and $P _ { \mathcal{C} } ^ { \# } ( n )$ have quite a different behaviour from that given by axiom $A ^ { \# }$. Here, although the objects in $\mathcal{C}$ may sometimes be rather complicated, the presently (as of 2000) known structure theorems for those objects often lead to a relatively simple estimation for $P _ { \mathcal{C} } ^ { \# } ( n )$ or $\pi _ { \mathcal{C} } ^ { \# } ( x ) = \sum _ { n \leq x } P _ { \mathcal{C} } ^ { \# } ( n )$. Surprisingly perhaps, it turns out that sharp asymptotic information can then be deduced about $G _ { \cal C } ^ { \# } ( n )$ or $N _ { \mathcal{C} } ^ { \# } ( x ) = \sum _ { n \leq x } G _ { \mathcal{C} } ^ { \# } ( n )$ by methods of classical-type arithmetical partition theory, which were initiated by G.H. Hardy and G. Ramanujan in 1917. These methods belong to a quite different branch of classical [[Analytic number theory|analytic number theory]] from those involved in the earlier discussion of axiom $A$.
  
On the basis of these new types of examples as motivation, one is led to investigations of an additive arithmetical semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050264.png" /> satisfying axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050266.png" />: There exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050267.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050268.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050269.png" /> (all depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050270.png" />) such that
+
On the basis of these new types of examples as motivation, one is led to investigations of an additive arithmetical semi-group $G$ satisfying axiom $C$: There exist constants $C > 0$, $\kappa > 0$ and $\nu$ (all depending on $G$) such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050271.png" /></td> </tr></table>
+
\begin{equation*} \pi _ { \mathcal{C} } ^ { \# } ( x ) \sim C x ^ { \kappa } ( \operatorname { log } x ) ^ { \nu } \text { as } x \rightarrow \infty. \end{equation*}
  
A simple example of axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050272.png" /> is provided when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050273.png" /> denotes either the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050274.png" /> of finite Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050275.png" />-groups (cf. also [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050276.png" />-group]]), or the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050277.png" /> of pseudo-metrizable finite topological spaces (cf. also [[Pseudo-metric space|Pseudo-metric space]]).
+
A simple example of axiom $C$ is provided when $\mathcal{C}$ denotes either the category $\mathcal{A} ( p )$ of finite Abelian $p$-groups (cf. also [[P-group|$p$-group]]), or the category $\mathcal{P}$ of pseudo-metrizable finite topological spaces (cf. also [[Pseudo-metric space|Pseudo-metric space]]).
  
Similar formulas hold for the categories of compact simply-connected Lie groups, or semi-simple finite-dimensional Lie algebras over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050278.png" /> of characteristic zero.
+
Similar formulas hold for the categories of compact simply-connected Lie groups, or semi-simple finite-dimensional Lie algebras over an algebraically closed field $F$ of characteristic zero.
  
Asymptotic deductions about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050279.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050280.png" />, subject to axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050281.png" />, could perhaps be referred to as  "inverse additive abstract prime number theorems" . Based on methods of generalized arithmetical partition theory, various theorems of this kind can be derived, as well as results about  "average values"  of arithmetical functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050282.png" />, and on asymptotic  "densities"  of certain subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050283.png" />, subject to axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050284.png" />.
+
Asymptotic deductions about $G ^ { \# } ( n )$ or $N _ { G } ^ { \# } ( x ) = \sum _ { n \leq x } G ^ { \# } ( n )$, subject to axiom $C$, could perhaps be referred to as  "inverse additive abstract prime number theorems" . Based on methods of generalized arithmetical partition theory, various theorems of this kind can be derived, as well as results about  "average values"  of arithmetical functions on $G$, and on asymptotic  "densities"  of certain subsets of $G$, subject to axiom $C$.
  
===Axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050285.png" />.===
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===Axiom $G_1$.===
Yet another natural class of additive arithmetical semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050286.png" /> is provided by those satisfying axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050288.png" />:  "Almost all"  elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050289.png" /> are prime, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050290.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050291.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050292.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050293.png" />.
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Yet another natural class of additive arithmetical semi-groups $G$ is provided by those satisfying axiom $G_1$:  "Almost all"  elements of $G$ are prime, in the sense that $G ^ { \# } ( n ) > 0$ for sufficiently large $n$, and $P ^ { \# } ( n ) \sim G ^ { \# } ( n )$ as $n \rightarrow \infty$.
  
It is known that various classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050294.png" /> of finite graphs define arithmetical semi-groups with this slightly surprising property. It is also known that, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050295.png" />, the multiplicative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050296.png" /> of all monic polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050297.png" /> indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050298.png" /> over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050299.png" /> has the property stipulated in axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130050/a130050300.png" />.
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It is known that various classes $\Gamma$ of finite graphs define arithmetical semi-groups with this slightly surprising property. It is also known that, when $k > 1$, the multiplicative semi-group $G _ { k , q }$ of all monic polynomials in $k$ indeterminates $X _ { 1 } , \ldots , X _ { k }$ over a finite field $\mathbf{F} _ { q }$ has the property stipulated in axiom $G_1$.
  
 
See [[Abstract prime number theory|Abstract prime number theory]] for a further discussion of arithmetical semi-groups and their corresponding abstract prime number theorems.
 
See [[Abstract prime number theory|Abstract prime number theory]] for a further discussion of arithmetical semi-groups and their corresponding abstract prime number theorems.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.-H. Indlekofer,  E. Manstavičius,  R. Warlimont,  "On a certain class of infinite products with an application to arithmetical semigroups"  ''Archiv Math.'' , '''56'''  (1991)  pp. 446–453</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Knopfmacher,  "Abstract analytic number theory" , North-Holland  (1975)  (Reprinted: Dover, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Knopfmacher,  "Analytic arithmetic of algebraic function fields" , M. Dekker  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.-B. Zhang,  "Elementary proofs of the abstract prime number theorem for algebraic function fields"  ''Trans. Amer. Math. Soc.'' , '''332'''  (1992)  pp. 923–937</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Knopfmacher,  W.-B. Zhang,  "Number theory arising from finite fields, analytic and probabilistic theory" , M. Dekker  (2001)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  K.-H. Indlekofer,  E. Manstavičius,  R. Warlimont,  "On a certain class of infinite products with an application to arithmetical semigroups"  ''Archiv Math.'' , '''56'''  (1991)  pp. 446–453</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Knopfmacher,  "Abstract analytic number theory" , North-Holland  (1975)  (Reprinted: Dover, 1990)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Knopfmacher,  "Analytic arithmetic of algebraic function fields" , M. Dekker  (1979)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  W.-B. Zhang,  "Elementary proofs of the abstract prime number theorem for algebraic function fields"  ''Trans. Amer. Math. Soc.'' , '''332'''  (1992)  pp. 923–937</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Knopfmacher,  W.-B. Zhang,  "Number theory arising from finite fields, analytic and probabilistic theory" , M. Dekker  (2001)</td></tr>
 +
</table>

Latest revision as of 07:41, 7 February 2024

The central concept in abstract analytic number theory is that of an arithmetical semi-group $G$ (defined below). It turns out that the study of such semi-groups and of (real- or complex-valued) functions on them makes it possible on the one hand to apply methods of classical analytic number theory in a unified way to a variety of asymptotic enumeration questions for isomorphism classes of different kinds of explicit mathematical objects. On the other hand, these procedures also lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration questions about the (mostly non-arithmetical) concrete types of mathematical objects just alluded to.


Arithmetical semi-groups

An arithmetical semi-group is, by definition, a commutative semi-group $G$ with identity element $1$, which contains a countable subset $P$ such that every element $a \ne 1$ in $G$ admits a unique factorization into a finite product of powers of elements of $P$, together with a real-valued mapping $\vert \cdot \vert$ on $G$ such that:

i) $|1| = 1$, $|p| > 1$ for $p \in P$;

ii) $|a b| = |a| \cdot |b|$ for all $a\,,b \in G$;

iii) the total number of elements $a$ with $|a| < x$ is finite, for each $x > 0$.

The elements of $P$ are called the primes of $G$, and $\vert \cdot \vert$ is called the norm mapping on $G$. It is obvious that, corresponding to any fixed $c>1$, the definition $\partial(a) = \log_c|a|$ yields a mapping $\partial$ on $G$ such that:

A) $\partial(1)=0$, $\partial(p)>0$ for $p \in P$;

B) $\partial(ab) = \partial(a) + \partial(b)$ for all $a\,,b \in G$;

C) the total number of elements $a$ with $\partial(a) \le x$ is finite, for each $x > 0$.

Conversely, any real-valued mapping $\partial$ with the properties A)–C) yields a norm on $G$, if one defines $|a| = c^{\partial(a)}$. In cases where such a mapping $\partial$ is of primary interest, $G$ together with $\partial$ is called an additive arithmetical semi-group, and one refers to $\partial$ as the degree mapping on $G$. In most concrete examples of interest, it turns out that the norm or degree mappings represent natural "size" or "dimension" measures which are integer-valued. With an eye to applications to natural examples there is therefore little loss in henceforth restricting attention to either a single integer-valued norm mapping $\vert\cdot\vert$, or a single integer-valued degree mapping $\partial$, on $G$. Depending on which case is being considered, special interest then attaches to the basic counting functions (for $n \in \mathbb{Z}$) $$ G(n) = \#\{ a \in G : |a| = n \} $$ $$ P(n) = \#\{ p \in P : |p| = n \} $$

(or $G^\sharp(n) = \#\{ a \in G : \partial(a) = n \}$, $P^\sharp(n) = \#\{ p \in P : \partial(p) = n \}$, in the additive case).

The prototype of all arithmetical semi-groups is of course the multiplicative semi-group $\mathbb{N}$ of all positive integers $\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime numbers $\{2,3,5,7,\ldots\}$. Here one may define the norm of an integer $|n|$ to be $n$, so that the number $\mathbb{N}(n) = 1$ for $n \ge 1$.

The asymptotic behaviour of $\pi(X) = \sum_{n \le x} P_{\mathbb{N}}(n)$ for large $x$ forms the content of the famous prime number theorem, which states that $$ \pi(x) \sim \frac{x}{\log x} \ \text{as}\ x \rightarrow \infty $$ (cf. also de la Vallée-Poussin theorem). A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semi-groups. For example, it is true for the multiplicative semi-group $G_K$ of all non-zero ideals in the ring $R = R(K)$ of all algebraic integers in a given algebraic number field $K$, with $|I| = \mathop{card}(R/I)$ for any non-zero ideal $I$ in $R$. Here, the prime ideals act as prime elements of the semi-group $G_K$.

A simple but nevertheless interesting example of an additive arithmetical semi-group is provided by the multiplicative semi-group $G-q$ of all monic polynomials in one indeterminate $X$ over a finite field $\mathbb{F}_q$ with $q$ elements, with $\partial a = \mathrm{deg}\, a$ and the set $P_q$ of prime elements represented by the irreducible polynomials (cf. also Irreducible polynomial). Here, $G_q^\sharp(n) = q^n$, and it can be proved that $$ P_q^\sharp (n) = \frac{1}{n} \sum_{r | n} \mu(r) q^{n/r} $$ where $\mu$ is the classical Möbius function on $\mathbb{N}$.

Up to isomorphism, $G_q$ is the simplest special case of the semi-group $G_R$ of all non-zero ideals in the ring $R = R(K)$ of all integral functions in an algebraic function field $K$ in one variable $K$ over $\mathbb{F}_q$.

Arithmetical categories of semi-groups.

Many interesting examples of concrete, but non-classical, arithmetical semi-groups can be found by considering certain specific classes of mathematical objects, such as groups, rings, topological spaces, and so on, together with appropriate "direct product" operations and isomorphism relations for those classes. It is convenient, though admittedly not quite precise, to temporarily ignore the corresponding morphisms and refer to such classes of objects as "categories" (cf. also Category).

Now consider some category $\mathcal{C}$ which admits a direct "product" (or "sum" ) operation $\times$ on its objects. Suppose that this operation $\times$ preserves $\mathcal{C}$-isomorphisms, is commutative and associative up to $\mathcal{C}$-isomorphism, and that $\mathcal{C}$ contains a "zero" object $0$ (unique up to $\mathcal{C}$-isomorphism) such that $A \times 0 \cong A$ for all objects $A$ in $\mathcal{C}$. Then suppose that a theorem of Krull–Schmidt type is valid for $\mathcal{C}$, i.e., suppose that every object $A$ can be expressed as a finite $\times$-product $A \cong P_1 \times \cdots \times P_k$ of objects $P_i$ that are indecomposable with respect to $\times$, in a way that is unique up to permutation of terms and $\mathcal{C}$-isomorphism. In most natural situations at least, one may reformulate these conditions on $\mathcal{C}$ by stating that the various isomorphism classes $\bar A$ of objects $A$ in $\mathcal{C}$ form a set $G_{\mathcal{C}}$ that is

i) a commutative semi-group with identity with respect to the multiplication operation $\bar A \times \bar B = \overline{A \times B}$;

ii) a semi-group with the unique factorization property with respect to the isomorphism classes of the indecomposable objects in $\mathcal{C}$.

For this reason, one may call the $\mathcal{C}$-isomorphism classes $\bar P_i$ of indecomposable objects $P$ the "primes" of $\mathcal{C}$ or $G_{\mathcal{C}}$.

In many interesting cases (some of which are illustrated below), the category $\mathcal{C}$ also admits a "norm" function $\vert \cdot \vert$ on objects which is invariant under $\mathcal{C}$-isomorphism and has the following properties:

i) $|0| = 1$, $|P| > 1$ for every indecomposable object $P$;

ii) $|A \times B| = |A|.|B|$ for all objects $A$, $B$;

iii) the total number of $\mathcal{C}$-isomorphism classes of objects $A$ of norm $|A| \le x$ is finite, for each real $x > 0$.

Obviously, in such circumstances, the definition $|\bar A| = |A|$ provides a norm function on $G_{\mathcal{C}}$ satisfying the required conditions for an arithmetical semi-group. For these reasons, a category $\mathcal{C}$ with such further properties may be called an arithmetical category.

Now consider some concrete illustrations for the above concepts, taken from [a2], [a3].

a) (Finite Abelian groups; cf. Abelian group.) One of the simplest non-trivial examples of an arithmetical category is provided by the category $\mathcal{A}$ of all finite Abelian groups, together with the usual direct product operation and the norm function $|A| = \mathrm{card}(A)$. Here, the Krull–Schmidt theorem reduces to the well-known fundamental theorem on finite Abelian groups, the indecomposable objects of this kind being simply the various cyclic groups $C_{p^n}$ of prime-power order $p^n$.

b) The category of all semi-simple associative rings of finite cardinality (cf. also Associative rings and algebras).

c) The category of all semi-simple finite-dimensional associate algebras over a given field $F$ (cf. also Associative rings and algebras; Semi-simple ring).

d) The category of all semi-simple finite-dimensional Lie algebras over a given field $F$.

e) The category of all compact simply-connected globally symmetric Riemannian manifolds (cf. also Globally symmetric Riemannian space).

f) The category $\mathcal{T}$ of topological spaces of finite cardinality with the property that a space $Y$ lies in $\mathcal{T}$ if and only if each connected component of $Y$ lies in $\mathcal{T}$.

Zeta-functions and enumeration problems.

For a given arithmetical semi-group $G$, information on the basic counting functions $G ( n )$, $P ( n )$ can often be obtained, algebraically or with the aid of analysis, via a certain series-production relation called the Euler product formula for $G$.

Indeed, ignoring questions of convergence for the moment, note that (by the unique factorization into prime elements of $G$) the series

\begin{equation*} \zeta _ { G } ( z ) = \sum _ { n = 1 } ^ { \infty } G ( n ) n ^ { - z } = \sum _ { a \in G } | a | ^ { - z } = \end{equation*}

\begin{equation*} = 1 + \sum | p _ { 1 } | ^ { - r _ { 1 } z } \ldots | p _ {m } | ^ { - r _ { m } z } = \end{equation*}

\begin{equation*} = \prod _ { p \in P } ( 1 + | p | ^ { - z } + | p | ^ { - 2 z } + \ldots ) = \end{equation*}

\begin{equation*} = \prod _ { p \in P } ( 1 - | p | ^ { - z } ) ^ { - 1 } = \end{equation*}

\begin{equation*} = \prod _ { m = 2 } ^ { \infty } ( 1 - m ^ { - z } ) ^ { - P ( m ) }. \end{equation*}

As a function of $z$, $\zeta _ { G } ( z )$ is called the zeta-function of $G$. If $G$ is an additive arithmetical semi-group with $| a | = c ^ { \partial ( a ) }$ for some integer $c > 1$, one may substitute the symbol $y$ for $c ^ { - z }$ and obtain the modified Euler product formula:

\begin{equation*} \sum _ { n = 0 } ^ { \infty } G ^ { \# } ( n ) y ^ { n } = \prod _ { m = 1 } ^ { \infty } ( 1 - y ^ { m } ) ^ { - P ^ { \# } ( m ) }; \end{equation*}

then $Z _ { G } ( y ) = \sum _ { n = 0 } ^ { \infty } G ^ { \# } ( n ) y ^ { n }$ is called the modified zeta-function (or generating function) of $G$.

Some explicit illustrations of zeta-functions and Euler products are given below.

The Riemann zeta-function.

For the basic semi-group $\mathbf{N}$ of positive integers, the zeta-function is $$ \zeta(z) = \sum_{n=1}^\infty n^{-z} \ ; $$ it is called the Riemann zeta-function, and the classical Euler product formula reads: $$ \zeta(z) = \prod_{\text{primes}\,p\,\text{of}\,\mathbf{N}} \left({ 1 - p^{-z} }\right)^{-1} \ . $$

The Dedekind zeta-function.

Let $G _ { K }$ denote the (above-mentioned) arithmetical semi-group of all non-zero "integral" ideals in a given algebraic number field $K$. The zeta-function for $G _ { K }$ is then

\begin{equation*} \zeta _ { K } ( z ) = \sum _ { I \in G _ { K } } | I | ^ { - z } = \sum _ { n = 1 } ^ { \infty } K ( n ) n ^ { - z }, \end{equation*}

where $K ( n )$ denotes the total number of ideals of norm $n$ in $G _ { K }$; it is known as the Dedekind zeta-function of $K$. (See also Zeta-function.)

Monic polynomials over a finite field.

For the additive arithmetical semi-group $G_q$ of all monic polynomials in one indeterminate $X$ over $\mathbf{F} _ { q }$ (see above), the generating function may be written as

\begin{equation*} Z _ { q } ( y ) = \sum _ { n = 0 } ^ { \infty } q ^ { n } y ^ { n } = ( 1 - q y ) ^ { - 1 }, \end{equation*}

and the above-mentioned explicit formula for $P _ { q } ^ { \# } ( n )$ can be deduced as an algebraic consequence of the Euler product for $G_q$.

Finite Abelian groups.

For the category $\mathcal{A}$ of all finite Abelian groups, the zeta-function may be written as

\begin{equation*} \zeta _ { A } ( z ) = \sum _ { n = 1 } ^ { \infty } a ( n ) n ^ { - z }, \end{equation*}

where $a ( n )$ denotes the total number of isomorphism classes of Abelian groups of order $n$. The discussion of "primes" in $\mathcal{A}$ given above shows that here the Euler product may be written as a double product

\begin{equation*} \zeta _ { A } ( z ) = \prod _ {\substack{ r \geq 1 \\ \text{primes } p \in \mathbf{N}}} \quad ( 1 - p ^ { - r z } ) ^ { - 1 } = \prod _ { r = 1 } ^ { \infty } \zeta ( r z ), \end{equation*}

by the Euler product formula for the Riemann zeta-function.

For the subcategory $\mathcal{A} ( p )$ of all finite Abelian $p$-groups, where $p$ is a fixed prime number (cf. also $p$-group), it is natural to regard $\mathcal{A} ( p )$ as an additive arithmetical category, with degree mapping defined by

\begin{equation*} \partial ( A ) = \operatorname { log } _ { p } \operatorname { card } ( A ). \end{equation*}

In that case, $\mathcal{A} ( p )$ has exactly one prime of degree $r$ for each $r = 1,2 , \dots$. Therefore the Euler product formula implies that $\mathcal{A} ( p )$ has the generating function

\begin{equation*} Z _ { \mathcal{A} ( p ) } ( y ) = \prod _ { r = 1 } ^ { \infty } ( 1 - y ^ { r } ) ^ { - 1 } = \sum _ { n = 0 } ^ { \infty } \mathbf{p} ( n ) y ^ { n }, \end{equation*}

where $\mathbf{ p} ( n ) = a ( p ^ { n } )$ is the total number of isomorphism classes of Abelian groups of degree $n$ in the above sense. In fact, for $n > 0$, $\mathbf{p} ( n )$ equals the total number of ways of partitioning $n$ into a sum of positive integers, which is also the number of pseudo-metrizable finite topological spaces of cardinality $n$ (see f) above). Thus, the corresponding latter category $\mathcal{P}$ (say) has the same generating function as $\mathcal{A} ( p )$.

Types of arithmetical semi-groups.

Bearing in mind the emphasis on concrete realizations of arithmetical semi-groups in a variety of areas of mathematics, it is reasonable to classify them and to base further investigations according to common features which may be exhibited by the initial enumeration theorems for particular sets of examples. In that way, further questions and enumeration problems may be investigated uniformly under suitable covering assumptions or "axioms" appropriate for particular natural sets of examples. On this basis, a small number of special types of arithmetical semi-groups have so far (2000) been found to predominate amongst natural concrete examples.

Classical and axiom-$A$ type semi-groups.

The strictly classical arithmetical semi-groups of analytic number theory are the multiplicative semi-group of all positive integers and the multiplicative semi-group of all non-zero ideals in the ring of all algebraic integers in a given algebraic number field (see above). For example, H. Weber and E. Landau proved theorems to the effect that

\begin{equation} \tag{a1} \sum _ { n \leq x } G _ { K } ( n ) = A _ { K } x + O ( x ^ { \eta_K} ) \text { as } x \rightarrow \infty, \end{equation}

where $G _ { K }$ is the semi-group of all "integral" ideals in a given algebraic number field $K$. Landau in particular used (a1) in order to extend many asymptotic results about arithmetical functions on $\mathbf{N}$ to similar functions on $G _ { K }$.

In quite a different direction, P. Erdős and G. Szekeres proved in 1934 for the category $\mathcal{A}$ of all finite Abelian groups that

\begin{equation} \tag{a2} \sum _ { n \leq x } a ( n ) = A _ { 1 } x + O ( \sqrt { x } ) \quad \text { as } x \rightarrow \infty, \end{equation}

where $A _ { 1 } = \prod _ { r \leq 2 } \zeta ( r ) = 2.29\dots$.

At a later stage, for the category $\mathcal{S}$ of semi-simple finite rings, I.G. Connell and J. Knopfmacher independently proved that

\begin{equation} \tag{a3} \sum _ { n \leq x } S ( n ) = A _ { 2 } x + O ( \sqrt { x } ) \quad \text { as } x \rightarrow \infty, \end{equation}

where $A _ { 2 } = \prod _ { rm ^ { 2 } \geq 2 } ^ { 2 } \zeta ( rm ^ { 2 } ) = 2.49 \dots$.

Strong concrete motivation was available for unifying certain further developments under the umbrella of general studies of an abstract arithmetical semi-group $G$ satisfying the so-called axiom $A$: There exist constants $A _ { G } > 0$, $\delta > 0$ and $\eta < \delta$ (all depending on $G$), such that

\begin{equation*} \sum _ { n \leq x } G ( n ) = A _ { G } x ^ { \delta } + O ( x ^ { \eta } ) \text { as } x \rightarrow \infty . \end{equation*}

Theorems based on the assumption of axiom $A$ often simultaneously generalize earlier results for $\mathbf{N}$, $G _ { K }$ and $G_{\mathcal{A}}$, and provide additional asymptotic enumeration theorems for a variety of arithmetical categories like $\mathcal{S}$ and many others.

Axiom $A ^ { \# }$ type semi-groups.

Consideration of the examples of multiplicative semi-groups of monic polynomials in one indeterminate, and also of enumeration theorems for some infinite families of explicit additive arithmetical categories connected with rings of integral functions in algebraic function fields over $\mathbf{F} _ { q }$ (cf. [a4], [a5]), provides a wealth of motivation for studying an abstract additive arithmetical semi-group $G$ satisfying axiom $A ^ { \# }$: There exist constants $A _ { G } > 0$, $q > 1$ and $\nu < 1$ (all depending on $G$) such that

\begin{equation*} G ^ { \# } ( n ) = A _ { G } q ^ { n } + O ( q ^ { \nu n } ) \text { as } n \rightarrow \infty. \end{equation*}

With this axiom as a basis instead of axiom $A$, problems similar to those outlined above may be investigated, with similar motivation to those stimulating the axiom-$A$ type studies. It then turns out that the ensuing results and methods of proof sometimes but not always possess parallels to those subject to axiom $A$.

A curious illustration of a non-parallel result arises with the abstract prime number theorem (or abstract prime element theorem) subject to axiom $A ^ { \# }$. In 1976, Knopfmacher derived such a theorem, on the initial foundation of some plausible-looking lemmas parallel to ones under axiom $A$. However, in 1989 and later, other authors independently found and then closed certain gaps in those lemmas. The combined efforts of various authors then led to a final theorem with two cases, depending on whether or not $Z _ { G } ( - q ^ { - 1 } ) = 0$; contributions to this were made by S.D. Cohen, K.-H. Indlekofer, E. Manstavičius, R. Warlimont and W.-B. Zhang (see e.g. [a1], [a4]).

A strange point about this result is that the case $Z _ { G } ( - q ^ { - 1 } ) \neq 0$ holds for all the natural examples which initially motivated axiom $A ^ { \# }$. Although ingenious examples in which $Z _ { G } ( - q ^ { - 1 } ) = 0$ have also been constructed, those found up to now might be viewed as somewhat pathological or contrived. Therefore, in terms of the "natural-example-based approach" to this subject outlined in the beginning, it would not be unreasonable to continue the present (2000) direction of investigation under the combined assumption of axiom $A ^ { \# }$ with the additional axiom

\begin{equation*} Z _ { G } ( - q ^ { - 1 } ) \neq 0. \end{equation*}

In fact (see e.g. [a3], [a5]) many consequences of axiom $A ^ { \# }$ are unrelated to the value of $Z _ { G } ( - q ^ { - 1 } )$, and so the simplifying additional axiom would only sometimes become relevant (but nevertheless reasonable to then assume at such a stage).

Axiom $C$.

The examples listed earlier included many involving an additive arithmetical category $\mathcal{C}$ for which $G _ { \cal C } ^ { \# } ( n )$ and $P _ { \mathcal{C} } ^ { \# } ( n )$ have quite a different behaviour from that given by axiom $A ^ { \# }$. Here, although the objects in $\mathcal{C}$ may sometimes be rather complicated, the presently (as of 2000) known structure theorems for those objects often lead to a relatively simple estimation for $P _ { \mathcal{C} } ^ { \# } ( n )$ or $\pi _ { \mathcal{C} } ^ { \# } ( x ) = \sum _ { n \leq x } P _ { \mathcal{C} } ^ { \# } ( n )$. Surprisingly perhaps, it turns out that sharp asymptotic information can then be deduced about $G _ { \cal C } ^ { \# } ( n )$ or $N _ { \mathcal{C} } ^ { \# } ( x ) = \sum _ { n \leq x } G _ { \mathcal{C} } ^ { \# } ( n )$ by methods of classical-type arithmetical partition theory, which were initiated by G.H. Hardy and G. Ramanujan in 1917. These methods belong to a quite different branch of classical analytic number theory from those involved in the earlier discussion of axiom $A$.

On the basis of these new types of examples as motivation, one is led to investigations of an additive arithmetical semi-group $G$ satisfying axiom $C$: There exist constants $C > 0$, $\kappa > 0$ and $\nu$ (all depending on $G$) such that

\begin{equation*} \pi _ { \mathcal{C} } ^ { \# } ( x ) \sim C x ^ { \kappa } ( \operatorname { log } x ) ^ { \nu } \text { as } x \rightarrow \infty. \end{equation*}

A simple example of axiom $C$ is provided when $\mathcal{C}$ denotes either the category $\mathcal{A} ( p )$ of finite Abelian $p$-groups (cf. also $p$-group), or the category $\mathcal{P}$ of pseudo-metrizable finite topological spaces (cf. also Pseudo-metric space).

Similar formulas hold for the categories of compact simply-connected Lie groups, or semi-simple finite-dimensional Lie algebras over an algebraically closed field $F$ of characteristic zero.

Asymptotic deductions about $G ^ { \# } ( n )$ or $N _ { G } ^ { \# } ( x ) = \sum _ { n \leq x } G ^ { \# } ( n )$, subject to axiom $C$, could perhaps be referred to as "inverse additive abstract prime number theorems" . Based on methods of generalized arithmetical partition theory, various theorems of this kind can be derived, as well as results about "average values" of arithmetical functions on $G$, and on asymptotic "densities" of certain subsets of $G$, subject to axiom $C$.

Axiom $G_1$.

Yet another natural class of additive arithmetical semi-groups $G$ is provided by those satisfying axiom $G_1$: "Almost all" elements of $G$ are prime, in the sense that $G ^ { \# } ( n ) > 0$ for sufficiently large $n$, and $P ^ { \# } ( n ) \sim G ^ { \# } ( n )$ as $n \rightarrow \infty$.

It is known that various classes $\Gamma$ of finite graphs define arithmetical semi-groups with this slightly surprising property. It is also known that, when $k > 1$, the multiplicative semi-group $G _ { k , q }$ of all monic polynomials in $k$ indeterminates $X _ { 1 } , \ldots , X _ { k }$ over a finite field $\mathbf{F} _ { q }$ has the property stipulated in axiom $G_1$.

See Abstract prime number theory for a further discussion of arithmetical semi-groups and their corresponding abstract prime number theorems.

References

[a1] K.-H. Indlekofer, E. Manstavičius, R. Warlimont, "On a certain class of infinite products with an application to arithmetical semigroups" Archiv Math. , 56 (1991) pp. 446–453
[a2] J. Knopfmacher, "Abstract analytic number theory" , North-Holland (1975) (Reprinted: Dover, 1990)
[a3] J. Knopfmacher, "Analytic arithmetic of algebraic function fields" , M. Dekker (1979)
[a4] W.-B. Zhang, "Elementary proofs of the abstract prime number theorem for algebraic function fields" Trans. Amer. Math. Soc. , 332 (1992) pp. 923–937
[a5] J. Knopfmacher, W.-B. Zhang, "Number theory arising from finite fields, analytic and probabilistic theory" , M. Dekker (2001)
How to Cite This Entry:
Abstract analytic number theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_analytic_number_theory&oldid=14794
This article was adapted from an original article by John Knopfmacher (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article