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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 $(\mathbb{Z}/n\mathbb{Z})^*$. They sketch a proof of the following Carlitz module analogue of this conjecture: Theorem. Let $A=\mathbb{F}_q[T]$, $C$ the $A$-Carlitz module, and $0\neq a\in A.$ For a monic polynomial $m\in A$, let $C(A/mA)$ and $\overline{a}$ be the reduction of $C$ and $a$ modulo $mA$ respectively. Let $f_a(m)$ be the monic generator of the ideal $\{f\in A,C_f(\overline{a})=\overline{0}\}$ on $C(A/mA)$. If $q\neq 2$, or $q=2$ and $a\neq 1,T$, or $1+T$, then for $\gamma\in\mathbb{R}$, we have $$\lim_{x\in\mathbb{N},x\to\infty}\frac{1}{q^x}\left|\left\{\deg m=x\quad\text{and}\quad\frac{\omega(f_a(m))-\frac12(\log x)^2}{(1/\sqrt{3})(\log x)^{3/2}}\leq \gamma\right\}\right|=G(\gamma),$$ where $\omega(f_a(m))$ is the number of distinct monic irreducible factors of $f_a(m)$, and $G(\gamma)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\gamma e^{-t^2/2}\,dt$ is the Gaussian normal distribution. For the entire collection see [Zbl 1142.11002].
11K65Arithmetic functions (probabilistic number theory)
11R58Arithmetic theory of algebraic function fields
11G09Drinfel’d modules, etc.