{"paper":{"title":"On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture","license":"","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alexei Tsygvintsev","submitted_at":"2004-12-15T11:25:32Z","abstract_excerpt":"We consider the limit periodic continued fractions of Stieltjes $$ \\frac{1}{1-} \\frac{g_1 z}{1-} \\frac{g_2(1-g_1) z}{1-} \\frac{g_3(1-g_2)z}{1-...,}, z\\in \\mathbb C, g_i\\in(0,1), \\lim\\limits_{i\\to \\infty} g_i=1/2, \\quad (1) $$ appearing as Shur--Wall $g$-fraction representations of certain analytic self maps of the unit disc $|w|< 1$, $w \\in \\mathbb C$. We precise the convergence behavior and prove the general convergence [2, p. 564 ] of (1) at the Runckel's points of the singular line $(1,+\\infty)$ It is shown that in some cases the convergence holds in the classical sense. As a result a count"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412298","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"}