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 . They sketch a proof of the following Carlitz module analogue of this conjecture:
Theorem. Let , the -Carlitz module, and For a monic polynomial , let and be the reduction of and modulo respectively. Let be the monic generator of the ideal on . If , or and , or , then for , we have
where is the number of distinct monic irreducible factors of , and is the Gaussian normal distribution.