Inverses of Hardy L-Functions

Abstract

Hardy conjectured that there exists an L‐function whose inverse is not asymptotic to any L‐function. We show that this is true of the L‐function log(logx).log(log(logx)).

Symbolic Asymptotics: Multiseries of Inverse Functions

“…For example Hardy (1911) states as a conjecture that there exist exp-log functions whose inverse is not asymptotically equivalent to an exp-log function. This conjecture was only proved recently (Shackell, 1993a;van den Dries et al, 1997;Van der Hoeven, 1997). We gave an algorithm for functional inversion of exp-log functions in terms of nested expansions in Salvy and Shackell (1992) In the present article, we develop an algorithm which produces another kind of expansion in the form of multiseries (precise definitions are given in Section 1).…”

“…It is a consequence of Liouville's theorem that in general this cannot be expected with base elements and coefficients which are exp-log functions. For instance, let f be an exp-log function whose inverse y is not asymptotic to an exp-log function (see Shackell, 1993a), then log f has an inverse exp(y) which cannot have a multiseries expansion since otherwise its logarithm would be asymptotic to an exp-log function.…”

Extended admissible functions and Gaussian limiting distributions

“…also Richardson et al [RSSVdH96]) in order to compute automatically an asymptotic expression for r n and, consequently, for the mean and variance if they exist. (The problem is that not every exp-log function has an inverse that is asymptotically equal to an exp-log functionl; see Shackell [Sha93]. )…”

Product Properties

“…The pressure gauge used by Shackell [4] during his studies of the drying process that would become known as lyophilization was a simple device known as a U tube. The U tube gauge, like that shown in Figure 11.4, is a device whose configuration is implied by its name.…”