2004/01/01 by Jonathan M. Borwein, Richard E. Crandall, G. J. Fee · 17 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Algebra and Geometry #Mathematics #Ramanujan's sum #Fraction (chemistry) #Algebraic number #Series (stratigraphy) #Rational number #Pure mathematics #Ramanujan tau function #Function (biology) #Rational function #Constant (computer programming) #Analytic number theory #Ramanujan theta function #Algebra over a field #Discrete mathematics #Mathematical analysis
paper · pdf · doi:10.1080/10586458.2004.10504540
published in Experimental Mathematics 13(3), 275-285 (Taylor & Francis)
openalex publication_date 2004/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
The Ramanujan AGM continued fraction is a construct enjoying attractive algebraic properties, such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function theory. But the fraction-also presents an intriguing computational challenge. Herein we show how to rapidly evaluate R for any triple of positive reals a, b,η. Even in the problematic scenario when a ≈ b certain transformations allow rapid evaluation. In this process we find, for example, that when a η = b η = a rational number, R η is essentially an L-series that can be cast as a finite sum of fundamental numbers. We ultimately exhibit an algorithm that yields D good digits of R in O(D) iterations where the implied big-O constant is independent of the positive-real triple a, b,η. Finally, we address the evidently profound theoretical and computational dilemmas attendant on complex parameters, indicating how one might extend the AGM relation for complex parameter domains.