# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Erdős-Kac theorem and its generalizations. (English) Zbl 1187.11024
De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 209-216 (2008).

The authors present a survey of the Erdös-Kac theorem and its various generalizations. In particular, they discuss an open conjecture of Erdös and Pomerance about the distribution of the number of distinct prime divisors of the order of a fixed integer in the multiplicative groups ${\left(ℤ/nℤ\right)}^{*}$. They sketch a proof of the following Carlitz module analogue of this conjecture:

Theorem. Let $A={𝔽}_{q}\left[T\right]$, $C$ the $A$-Carlitz module, and $0\ne a\in A·$ For a monic polynomial $m\in A$, let $C\left(A/mA\right)$ and $\overline{a}$ be the reduction of $C$ and $a$ modulo $mA$ respectively. Let ${f}_{a}\left(m\right)$ be the monic generator of the ideal $\left\{f\in A,{C}_{f}\left(\overline{a}\right)=\overline{0}\right\}$ on $C\left(A/mA\right)$. If $q\ne 2$, or $q=2$ and $a\ne 1,T$, or $1+T$, then for $\gamma \in ℝ$, we have

$\underset{x\in ℕ,x\to \infty }{lim}\frac{1}{{q}^{x}}\left|\left\{degm=x\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\frac{\omega \left({f}_{a}\left(m\right)\right)-\frac{1}{2}{\left(logx\right)}^{2}}{\left(1/\sqrt{3}\right){\left(logx\right)}^{3/2}}\le \gamma \right\}\right|=G\left(\gamma \right),$

where $\omega \left({f}_{a}\left(m\right)\right)$ is the number of distinct monic irreducible factors of ${f}_{a}\left(m\right)$, and $G\left(\gamma \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\gamma }{e}^{-{t}^{2}/2}\phantom{\rule{0.166667em}{0ex}}dt$ is the Gaussian normal distribution.

##### MSC:
 11K65 Arithmetic functions (probabilistic number theory) 11R58 Arithmetic theory of algebraic function fields 11G09 Drinfel’d modules, etc.
##### Keywords:
number of distinct prime divisors