Explicit Tauberian estimates for functions with positive coefficients
type: Item
instance of: scholarly article
title: Explicit Tauberian estimates for functions with positive coefficients
zbMATH Open document ID: 0763.40006
DOI: 10.1016/0377-0427(92)90248-V
published in: Journal of Computational and Applied Mathematics
publication date: 1992-10-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 40E05
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 69616
zbMATH Keywords: Tauberian theorem
zbMATH Keywords: Laplace-Stieltjes transform
zbMATH Keywords: Rankin's method
author: Andrew Odlyzko
reviewed by: D. C. Russell
MaRDI profile type: Publication
cites work: Q4146776
cites work: Q5335427
cites work: A General Asymptotic Formula for Partition Functions
cites work: Q5514677
cites work: On the packing densities of superballs and other bodies
cites work: Tauberian remainder theorems
cites work: On Integers Free of Large Prime Factors
cites work: Lattice points in high-dimensional spheres
cites work: Q4732505
cites work: Q3871318
cites work: The Difference between Consecutive Prime Numbers
Recommended article: On the Local Behavior of Ψ(x, y)
Recommended article: Q3490087
Recommended article: On Integers Free of Large Prime Factors
Recommended article: Unbalanced quadratic residues and non-residues
Recommended article: Arithmetic properties of the Ramanujan function
Recommended article: Q4052219
Recommended article: A Differential Delay Equation Arising from the Sieve of Eratosthenes
Recommended article: The difference between consecutive prime numbers. II.
Recommended article: Q5471893
Recommended article: LOCAL DENSITIES OVER INTEGERS FREE OF LARGE PRIME FACTORS
description: scientific article; zbMATH DE number 69616