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 $f:[1,\infty )\to \mathbb{R}$ 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.

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