Non-polar singularities of local zeta functions in some smooth case

TL;DR

This paper investigates the analytic continuation of local zeta functions associated with smooth functions, revealing the existence of non-polar singularities and explicitly computing their asymptotic behaviors.

Contribution

It provides a detailed analysis of non-polar singularities in local zeta functions for smooth functions, extending understanding beyond real analytic cases.

Findings

Local zeta functions can have non-polar singularities.

Asymptotic limits at singularities are explicitly computed.

Singularities differ from poles in certain smooth cases.

Abstract

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (non-real analytic) smooth functions is precisely investigated. Indeed, asymptotic limits of the respective local zeta functions at some singularities in one direction are explicitly computed. Surprisingly, it follows from these behaviors that these local zeta functions have singularities different from poles.