Tauberian theorems concerning Laplace transforms and Dirichlet series (Q1108495)

It is shown, inter alia, that under certain general Tauberian conditions the asymptotic relationship \[ \int^{\infty}_{0}s(v)e^{- vx}dA(v)\sim \ell \int^{\infty}_{0}e^{-vx}dA(v)\quad as\quad x\to 0+ \] between two Laplace-Stieltjes transforms implies that s(v)\(\to \ell\) as \(v\to \infty\).