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.
Categories: math.DS math.CV