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

Jason P. Bell ¹   ; Jeffrey C. Lagarias ²

¹ Department of Mathematics University of Waterloo Waterloo, Ontario, Canada
² Department of Mathematics University of Michigan Ann Arbor, MI 48109-1043, U.S.A.

Acta Arithmetica, Tome 170 (2015) no. 2, pp. 101-120

Résumé

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$, resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1\, ({\rm mod}\, 6)$. The map $T_k(\cdot)$ sends integers to integers; for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0,\, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ to 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 $m=1,2, 4, 8$.

Détail

DOI : 10.4064/aa170-2-1

Keywords: function sends resp according odd resp even where equiv mod map cdot sends integers integers rightarrow mean forward orbit under iteration cdot consider generating functions sum rightarrow which holomorphic unit disk sufficient conditions functions have unit circle natural boundary analytic continuation function these conditions has unit circle natural boundary except possibly conjecture equivalent assertion rational function remaining values