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

Acta Arithmetica (2015)

• Volume: 170, Issue: 2, page 101-120
• ISSN: 0065-1036

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Abstract

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The 3x+k function ${T}_{k}\left(n\right)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map ${T}_{k}\left(·\right)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of ${T}_{k}\left(·\right)$. We consider the generating functions ${f}_{k,m}\left(z\right)={\sum }_{n>0,n\to m}{z}^{n}$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions ${f}_{k,m}\left(z\right)$ 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 ≥1 to show that ${f}_{1,m}\left(z\right)$ 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 ${f}_{1,m}\left(z\right)$ is a rational function of z for the remaining values m = 1,2,4,8.

How to cite

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Jason P. Bell, and Jeffrey C. Lagarias. "3x+1 inverse orbit generating functions almost always have natural boundaries." Acta Arithmetica 170.2 (2015): 101-120. <http://eudml.org/doc/279152>.

@article{JasonP2015,
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 ≡ ±1 (mod 6). The map $T_k(·)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_k(·)$. We consider the generating functions $f_\{k,m\}(z) = ∑_\{n>0, n → m\} z^\{n\}$, 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 ≥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.},
author = {Jason P. Bell, Jeffrey C. Lagarias},
journal = {Acta Arithmetica},
keywords = {Collatz problem; Skolem-Mahler-Lech theorem; difference equations; discrete dynamical systems},
language = {eng},
number = {2},
pages = {101-120},
title = {3x+1 inverse orbit generating functions almost always have natural boundaries},
url = {http://eudml.org/doc/279152},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Jason P. Bell
AU - Jeffrey C. Lagarias
TI - 3x+1 inverse orbit generating functions almost always have natural boundaries
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 2
SP - 101
EP - 120
AB - 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 ≡ ±1 (mod 6). The map $T_k(·)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_k(·)$. We consider the generating functions $f_{k,m}(z) = ∑_{n>0, n → m} z^{n}$, 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 ≥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.
LA - eng
KW - Collatz problem; Skolem-Mahler-Lech theorem; difference equations; discrete dynamical systems
UR - http://eudml.org/doc/279152
ER -

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