On convergence of generalized continued fractions and Ramanujan's conjecture
| dc.creator | Glutsyuk, A. A. | |
| dc.date | 2004-12-22 | |
| dc.date | 2004-12-23 | |
| dc.date.accessioned | 2026-07-07T05:15:34Z | |
| dc.date.available | 2026-07-07T05:15:34Z | |
| dc.description | We consider continued fractions \frac{-a_1}{1-\frac{a_2}{1-\frac{a_3}{1-...}}} \label{fr} with real coefficients $a_i$ converging to a limit $a$. S.Ramanujan had stated the theorem (see [ABJL], p.38) saying that if $a\neq\frac14$, then the fraction converges if and only if $a<\frac14$. The statement of convergence was proved in [V] for complex $a_i$ converging to $a\in\mathbb C\setminus[\frac14,+\infty)$ (see also [P]). J.Gill [G] proved the divergence of (\ref{fr}) under the assumption that $a_i\to a>\frac14$ fast enough, more precisely, whenever \sum_i|a_i-a|<\infty.\label{gill} The Ramanujan conjecture saying that (\ref{fr}) diverges always whenever $a_i\to a>\frac14$ remained up to now an open question. In the present paper we disprove it. We show (Theorem \ref{th1}) that for any $a>\frac14$ there exists a real sequence $a_i\to a$ such that (\ref{fr}) converges. Moreover, we show (Theorem \ref{go}) that Gill's sufficient divergence condition (\ref{gill}) is the optimal condition on the speed of convergence of the $a_i$'s. | |
| dc.description | Submitted to C.R.A.S. on December 23, 2004 | |
| dc.identifier | https://arxiv.org/abs/math/0412445 | |
| dc.identifier | http://arxiv.org/abs/math/0412445 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73673 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Complex Variables | |
| dc.subject | 37F99 | |
| dc.title | On convergence of generalized continued fractions and Ramanujan's conjecture | |
| dc.type | text |