Inverse Functions

Abstract

Inverse functions have traditionally been regarded as problematic in asymptotics. For example, until recently it was unknown whether the inverse of an exp-log function was necessarily asymptotic to an exp-log function. Thus in [40], Hardy said ‘whether or not it is true that, given an L-function φ and its inverse $\bar \emptyset$, there must be an L-function ψ such that $\bar \emptyset$ ~ ψ I cannot say; ...I am very doubtful whether this is so’. Similarly [30] treats a number of examples and remarks on the difficulty of some of them. In fact once one has an understanding of scales and the mechanisms of nested forms and multiseries, inverse function are reasonably tractable. In particular, Hardy’s conjecture was established in [98] by showing that the inverse of the function log log x · log log log x is not asymptotic to any L-function. Subsequently it was shown in [113], and independently in [114], that the same is true of the inverse of log x · log log x,which was suggested as a possible counter-example by Hardy in [40].