lambda(lambda(n)): A case study in analytic number theory

A 2005 result of Carl Pomerance and myself identifies the normal order (that is, the asymptotic size for 100% of integers) of the twice-iterated Carmichael lambda-function ?(?(n)), a function that arises when considering an exponential pseudorandom number generator xk+1 = xkC (mod n). Often I suppress most of the technical details of the multi-stage proof in talks on this topic; however, today I will use the result as an excuse to point out the techniques involved, techniques that might be labeled “Erdos mathematics” — elementary yet involved and (in his case) inspired calculations.