On Ikehara type Tauberian theorems with O(xγ) remainders

Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for f : [ 1 , ∞ ) → R non-negative and non-decreasing we prove f ( x ) - x = O ( x γ ) with γ < 1 under certain assumptions on f . We state a conje...

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Bibliographic Details
Published in:Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2018-04, Vol.88 (1), p.209-216
Main Author: Müger, Michael
Format: Article
Language:English
Subjects:
Algebra
Differential Geometry
Geometry
Mathematics
Mathematics and Statistics
Number Theory
Theorems
Topology
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Summary:Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for f : [ 1 , ∞ ) → R non-negative and non-decreasing we prove f ( x ) - x = O ( x γ ) with γ < 1 under certain assumptions on f . We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.
ISSN:0025-5858
1865-8784
DOI:10.1007/s12188-017-0187-0