{"paper":{"title":"$3x+1$ inverse orbit generating functions almost always have natural boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CV","authors_text":"Jason P. Bell, Jeffrey C. Lagarias","submitted_at":"2014-08-28T23:34:00Z","abstract_excerpt":"The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \\equiv \\pm 1~(\\bmod \\, 6)$. The map $T_k(\\cdot)$ sends integers to integers, and 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 $f_{k, m}(z)$ have the unit circle $\\{|z|=1\\}$ as a natural boundary to analytic continuation. For the $3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6884","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}