A remark on the Wiener-Ikehara Tauberian theorem

TL;DR

This paper discusses how Kable's extension of the Wiener-Ikehara Tauberian theorem can be adapted to handle Dirichlet series with poles of fractional order, broadening its applicability.

Contribution

It demonstrates that the proof technique for Kable's extension applies to poles of order l/m, extending previous results for simple poles.

Findings

Extension of the Wiener-Ikehara theorem to fractional pole orders

Simplified adaptation of Kable's proof for new cases

Broader applicability of Tauberian theorems in number theory

Abstract

In this paper we point out that the proof of Kable's extension of the Wiener-Ikehara Tauberian theorem can be applied to the case where the Dirichlet series has a pole of order "$l / m$" without much modification (Kable proved the case $l = 1$).