On convergence of generalized continued fractions and Ramanujan's conjecture

Abstract

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 $a\ne \frac{1}{4}$, then the fraction converges if and only if $a<\frac{1}{4}$. The statement of convergence was proved in [V] for complex $a_i$ converging to $a\in \mathbb{C}\setminus [\frac{1}{4},+\infty )$ (see also [P]). J.Gill [G] proved the divergence of (\ref{fr}) under the assumption that $a_i\to a>\frac{1}{4}$ fast enough, more precisely, whenever \sum_i|a_i-a|<\infty.\label{gill} The Ramanujan conjecture saying that (\ref{fr}) diverges always whenever $a_i\to a>\frac{1}{4}$ remained up to now an open question. In the present paper we disprove it. We show (Theorem \ref{th1}) that for any $a>\frac{1}{4}$ 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.

Submitted to C.R.A.S. on December 23, 2004