Modular forms and Eisenstein's continued fractions
نویسندگان
چکیده
منابع مشابه
Continued Fractions and Modular Forms
This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coeecients of a continued fraction, thus solving the longstanding open problem of their limit law.
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Found in the collected works of Eisenstein are twenty continued fraction expansions. The expansions have since emerged in the literature in various forms, although a complete historical account and self-contained treatment has not been given. We provide one here, motivated by the fact that these expansions give continued fraction expansions for modular forms. Eisenstein himself did not record p...
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Abstract. We consider certain probability problems which are naturally related to integer partitions. We show that the corresponding probabilities are values of classical modular forms. Thanks to this connection, we then show that certain ratios of probabilities are specializations of the Rogers-Ramanujan, and RamanujanSelbergGordon-Göllnitz continued fractions. One particular evaluation depend...
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It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the spec...
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Using the techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of consecutive convergents. We then study some averages related to modular symbols. §0. Introduction and summary 0.1. Continued fractions. We start by fixing the nota...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2006
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2005.06.001