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## Acta Arithmetica

2015 | 170 | 2 | 101-120
Tytuł artykułu

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

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EN
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EN
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.
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Tom
Numer
Strony
101-120
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Daty
wydano
2015
Twórcy
autor
• Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada
autor
• Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A.
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