Meromorphic continuation and non-polar singularities of local zeta functions in some smooth cases

TL;DR

This paper investigates the meromorphic continuation and singularity types of local zeta functions associated with certain smooth functions, revealing non-polar singularities and asymptotic behaviors that differ from classical real analytic cases.

Contribution

It extends the understanding of local zeta functions to specific smooth functions, identifying non-polar singularities and establishing optimal lower bounds for their meromorphic continuation.

Findings

Local zeta functions exhibit non-polar singularities in certain smooth cases.

Asymptotic limits of local zeta functions at singularities are characterized.

Optimal lower estimates for meromorphic continuation are established for specific smooth functions.

Abstract

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the hole complex plane. In this paper, certain cases of specific (non-real analytic) smooth functions are precisely investigated. In particular, we give asymptotic limits of local zeta functions at some singularities along one direction. It follows from the behaviors that these local zeta functions have singularities different from poles. Then we show the optimality of the lower estimates of a certain quantity concerning with meromorphic continuation of local zeta functions in the case of all smooth functions expressed as $u(x,y)x^a y^b +$ flat function, where $u(0,0)\neq 0$ and $a,b$ are nonnegative integers satisfying $a\neq b$.