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

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.