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

TL;DR

This paper investigates power law remainder terms in Ikehara's Tauberian theorem, establishing conditions under which the function's deviation from linearity is bounded by a power of x, motivated by applications in analytic number theory.

Contribution

It provides new conditions for Ikehara's theorem with $O(x^eta)$ remainders and discusses the optimality of these assumptions.

Findings

Proves $f(x)-x=O(x^eta)$ under specific conditions

States a conjecture on the minimal assumptions needed

Shows limitations on improving the remainder bounds

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.