On convergence of generalized continued fractions and Ramanujan's conjecture

Alexey A. Glutsyuk

doi:10.1016/j.crma.2005.08.001

Dynamical Systems

On convergence of generalized continued fractions and Ramanujan's conjecture

Presented by: Étienne Ghys

Alexey A. Glutsyuk ¹

¹ CNRS, unité de mathématiques pures et appliquées, M.R., École normale supérieure de Lyon, 46, allée d'Italie, 69364 Lyon cedex 07, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 427-432.

Abstract
Résumé

We consider continued fractions

\frac{- a_{1}}{1 - \frac{a_{2}}{1 - \frac{a_{3}}{1 - \dots}}}

(CF)

with real coefficients

a_{i}

converging to a limit a. Ramanujan claimed that if

a \neq \frac{1}{4}

, then the fraction converges if and only if

a < \frac{1}{4}

. The statement of convergence was proved by Van Vleck in 1904 for complex

a_{i}

converging to

a \in C ∖ [\frac{1}{4}, + \infty)

. Gill proved the divergence of (CF) under the assumption that

a_{i} \to a > \frac{1}{4}

fast enough, more precisely, whenever

\sum_{i} | a_{i} - a | < \infty

. The Ramanujan's conjecture saying that (CF) always diverges whenever

a_{i} \to a > \frac{1}{4}

remained, up to now, an open question. In the present Note we disprove it. We show that for any

a > \frac{1}{4}

there exists a real sequence

a_{i} \to a

such that (CF) converges. Moreover, we show that Gill's sufficient divergence condition is the optimal condition on the speed of convergence of the

a_{i}

's.

Nous considérons une fraction continue

\frac{- a_{1}}{1 - \frac{a_{2}}{1 - \frac{a_{3}}{1 - \dots}}}

(FC)

à coéfficients réels

a_{i} \to a

. Ramanujan a affirmé, que si

a \neq \frac{1}{4}

, alors la fraction converge, si et seulement si

a < \frac{1}{4}

. La convergence a été démontrée par Van Vleck en 1904 pour

a_{i}

complexes convergeant vers un

a \in C ∖ [\frac{1}{4}, + \infty)

. Gill a démontré (en 1973), que la fraction diverge, si

a_{i} \to a > \frac{1}{4}

assez vite, plus précisement, si

\sum_{i} | a_{i} - a | < \infty

.

La conjecture de Ramanujan disant que la fraction diverge toujours, quand $a > \frac{1}{4}$ , restait ouverte jusqu'au présent. Nous montrons, qu'elle est fausse : pour tout $a > \frac{1}{4}$ il existe une suite réelle $a_{i} \to a$ telle que la fraction converge. Nous montrons aussi, que la condition précedante de Gill, qui est suffisante pour que la fraction diverge, est celle optimale sur la vitesse de convergence des $a_{i}$ .

Received: 2004-12-23
Accepted: 2005-07-19
Published online: 2005-09-13

DOI: 10.1016/j.crma.2005.08.001

Author's affiliations:

Alexey A. Glutsyuk ¹

¹ CNRS, unité de mathématiques pures et appliquées, M.R., École normale supérieure de Lyon, 46, allée d'Italie, 69364 Lyon cedex 07, France

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     title = {On convergence of generalized continued fractions and {Ramanujan's} conjecture},
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     pages = {427--432},
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     year = {2005},
     doi = {10.1016/j.crma.2005.08.001},
     language = {en},
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Alexey A. Glutsyuk. On convergence of generalized continued fractions and Ramanujan's conjecture. Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 427-432. doi : 10.1016/j.crma.2005.08.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.08.001/

References
Cited by

[1] G.E. Andrews; B.C. Berndt; L. Jacobsen; R.L. Lamphere The continued fractions found in the unorganized portions of Ramanujan's notebooks, Mem. Amer. Math. Soc., Volume 99 (1992) no. 477

[2] J. Gill Infinite composition of Möbius transformations, Trans. Amer. Math. Soc., Volume 176 (1973), pp. 479-487

[3] L. Lorentzen On the convergence of limit periodic continued fractions $K (a_{n} / 1)$ where $a_{n} \to - \frac{1}{4}$ . IV, Constr. Approx., Volume 18 (2002) no. 1, pp. 1-17

[4] O. Perron Die Lehre von den Kettenbrüchen, Band 2, B.G. Teubner, Stuttgart, 1957

[5] A. Tsygvintsev, On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture, submitted for publication

[6] E.B. Van Vleck On the convergence of algebraic continued fractions whose coefficients have limiting values, Trans. Amer. Math. Soc., Volume 5 (1904), pp. 253-262

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