This paper speeds up Brent's algorithms for various high-precision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square root; roughly 9 to compute both a square root and a reciprocal square root; and roughly 10:4 to compute an exponential. The same ideas apply to approximate computations in R, Q p, etc.

This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.

This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFT-based power-series exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.

We consider Ramanujan's contribution to formulas for Euler's constant fl. For example, in his second notebook Ramanujan states that (in modern notation) 1 X k=1 (\Gamma1) k\Gamma1 nk ` x k k! ' n = ln x + fl + o(1) as x ! 1. This is known to be correct for the case n = 1, but incorrect for n ? 2. We consider the case n = 2. We also suggest a different, correct generalization of the case n = 1. 1.