On the Ramanujan AGM Fraction, I: The Real-Parameter Case

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 aη = bη = a rational number, Rη is essentially an Lseries 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.