AMS Nos: 11J70,11Y65
Key Words Continued fractions, computational number theory

Abstract. 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 rapidluy evaluate R for any triple of positive reals a, b, eta, the problemtic scenario being when a = b, although even in such cases certain transformations allow rapid evaluation. In this process we find, for example, that when a=b=rational, Reta is essentially an L-series that can be cast therefore 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, eta. Finally, we address the evidently profound theoretical and computational dilemmas that arise when the parameters are allowe to become complex, finding means to extend the AGM relation for complex parameter domains.