$3x+1$ inverse orbit generating functions almost always have natural boundaries

Abstract

The $3x+k$ function $T_k(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k\equiv \pm 1(\mathrm{mod}6)$. The map $T_k(\cdot )$ sends integers to integers, and for $m\ge 1$ let $n\to m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot ).$ We consider the generating functions $f_{k,m}(z)={\sum }_{n>0,n\to m}{z}^n,$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k,m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m\ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m=1,2,4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1,m}(z)$ is a rational function of $z$ for the remaining values $m=1,2,4,8$.

15 pages