One version of the Ingham-Karamata theorem states that for each slowly oscillating function whose Laplace transform admits an analytic continuation beyond the line must obey the asymptotic law . This theorem is a cornerstone in Tauberian theory and has plenty of applications in number theory; one of the quickest proofs of the Prime Number Theorem passes through this theorem.
We shall show that the decay rate in the Ingham-Karamata theorem is optimal even if one assumes analytic continuation of the Laplace transform up to a larger halfplane. The attractive proof is based on the open mapping theorem.
