Note on the absence of remainders in the Wiener-Ikehara theorem

TL;DR

This paper demonstrates the limitations of improving remainders in the Wiener-Ikehara and Ingham-Karamata theorems, even under assumptions of analytic continuation, highlighting fundamental constraints in these classical results.

Contribution

It establishes that better remainders cannot be achieved in these theorems despite assumptions of analytic continuation, clarifying inherent limitations.

Findings

No improved remainders beyond classical ones are possible.

Analytic continuation assumptions do not enhance the remainder estimates.

Results apply to both Wiener-Ikehara and Ingham-Karamata theorems.

Abstract

We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.