On Tauberian Theorems

A typical Tauberian theorem deduces an asymptotic for the partial sums of a sequence of non-negative real numbers from analytic properties of an associated Dirichlet series. Tauberian theorems appear in a tremendous variety of applications, and are so “classical” that sometimes practitioners omit a reference. But there are reasons for caution. We will survey two types of Tauberian theorems, and mention counterexamples that illuminate that neither theorem can reach an essentially stronger conclusion unless its hypothesis is strengthened.