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

Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for $f:[1,\mathrm{\infty })\to \mathbb{R}$ 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.