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

Tom 170 / 2015

Jason P. Bell, Jeffrey C. Lagarias Acta Arithmetica 170 (2015), 101-120 MSC: Primary 30B40; Secondary 11B83, 11K31, 26A18, 30B10, 37A45. DOI: 10.4064/aa170-2-1


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 $m=1,2, 4, 8$.


  • Jason P. BellDepartment of Mathematics
    University of Waterloo
    Waterloo, Ontario, Canada
  • Jeffrey C. LagariasDepartment of Mathematics
    University of Michigan
    Ann Arbor, MI 48109-1043, U.S.A.

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