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

Michael Müger

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arXiv · 1612.01779

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

Michael Müger

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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.

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Michael Müger. 2016-12-06. On Ikehara type Tauberian theorems with $O(x^\gamma)$ remainders. https://arxiv.org/abs/1612.01779

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