Explicit Tauberian estimates for functions with positive coefficients (Q1195419)

Given the Laplace-Stieltjes transform \(F(s):=\int_ 0^ \infty e^{- sx} d\mu(x)\), with \(\mu(\cdot)\) of locally bounded variation, related Tauberian theorems obtain information about \(\mu(\cdot)\) from information about \(F(\cdot)\). For example, if \(\mu(\cdot)\) is nondecreasing with \(\mu(0)=0\), and if the integral defining \(F(s)\) converges for all \(s>0\), then for any \(y>0\) and any \(s>0\), we have \(\mu(y)\leq e^{sy}F(s)\). In the case where \(\mu(x):=\sum_{0\leq\log n\leq x}a_ n\), we get the Dirichlet series \(F(s)=\sum_{n\geq 1} a_ n n^{-s}\), and the above upper bound for \(\mu(y)\) was used by \textit{R. A. Rankin} [J. Lond. Math. Soc. 13, 242-247 (1938; Zbl 0019.39403)] in the study of integers without large prime factors. In the present paper, for nondecreasing \(\mu(\cdot)\) with \(\mu(0)=0\) and \(\mu(x)\to\infty\) as \(x\to\infty\), the author uses a ``saddle-point method'' to give sufficient conditions under which an explicit lower bound for \(\mu(y)-\mu(y-c)\) (for a suitable constant \(c\)) can be specified. Among interesting applications are results (some of them known) on partitions, lattice points in superballs, and integers without large prime factors.