Lecture Notes in Mathematics. 1681. Berlin: Springer. vii, 158 p. DM 45.00; öS 329.00; sFr. 41.50; £ 17.50; $ 33.00 (1998).

One of the most fascinating mathematical problems is the “Collatz”-problem (or “$3x+1$”- or “Hasse”- or “Syracuse”- or “Kakutani”-problem), which is to prove that for every $n \in \Bbb N$ there exists a $k$ with $C^{(k)}(n)=1$ where the Collatz function $C(n)$ takes odd numbers $n$ to $3n+1$ and even numbers $n$ to $n/2$.
In the volume under review, the author gives a complete and competent treatise on the still open problem, although it was attacked by a surprisingly broad variety of different mathematical methods.
The five chapters are organized as follows: Chapter I: Introduction, including probability analysis, continued fractions, formal languages; Chapter II: Discrete dynamical systems on the natural numbers and the Collatz graph, density estimates; Chapter III: The $3$-adic averages of counting functions; Chapter IV: An asymptotically homogeneous Markov chain; Chapter V: A reduction theorem.
The origin dates back to the 1930’s, but in the last twenty years there have been a growing number of papers published on this topic; the bibliography contains about 120 items. As the original problem seems to be intractable, there are a lot of interesting generalizations or weaker statements implied by the conjecture.
The note is very well written. It is an excellent work for all mathematicians who deal with recursive number-theoretic functions.