On Ikehara type Tauberian theorems with $O(x^\gamma)$ remainders

Michael Müger

On Ikehara type Tauberian theorems with $O(x^\gamma)$ remainders

Classical Analysis and ODEs 2019-10-29 v2

Authors: Michael Müger

Abstract

Motivated by analytic number theory, we explore remainder versions of Ikehara's Tauberian theorem yielding power law remainder terms. More precisely, for $f:[1,\infty)\rightarrow{\mathbb R}$ non-negative and non-decreasing we prove $f(x)-x=O(x^\gamma)$ with $\gamma<1$ under certain assumptions on $f$ . We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.

Keywords

Ikehara Tauberian theorem power law remainder analytic number theory Tauberian theorem remainder term

Cite

@article{arxiv.1612.01779,
  title  = {On Ikehara type Tauberian theorems with $O(x^\gamma)$ remainders},
  author = {Michael Müger},
  journal= {arXiv preprint arXiv:1612.01779},
  year   = {2019}
}

Comments

A serious mistake in the first version was corrected and everything rewritten, including new material and a conjecture. 6 pages

On Ikehara type Tauberian theorems with $O(x^\gamma)$ remainders

Abstract

Keywords

Cite

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