  We present here an overview of the history, applications and important
properties of a function which we refer to as the Levy integral. For certain
values of its characteristic parameter the Levy integral defines the symmetric
Levy stable probability density function. As we discuss however the Levy
integral has applications to a number of other fields besides probability,
including random matrix theory, number theory and asymptotics beyond all
orders. We exhibit a direct relationship between the Levy integral and a number
theoretic series which we refer to as the generalised Euler-Jacobi series. The
complete asymptotic expansions for all natural values of its parameter are
presented, and in particular it is pointed out that the intricate exponentially
small series become dominant for certain parameter values.
