Meromorphy of local zeta functions in smooth model cases

TL;DR

This paper studies the meromorphic extension of local zeta functions for smooth functions with specific flatness properties, revealing new phenomena and precise pole structures in certain cases.

Contribution

It classifies flat functions into four types and analyzes the meromorphic extension of local zeta functions for each, uncovering novel behaviors in the smooth case.

Findings

Local zeta functions extend meromorphically in certain smooth cases.

Poles are contained in specific rational sets depending on parameters.

New phenomena occur when the function's flatness type influences extension properties.

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. But, in the case of general ($C^{\infty}$) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as $u(x,y)x^a y^b +$ flat function, where $u(0,0)\neq 0$ and $a,b$ are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each cases. Our results show new interesting phenomena in one of these cases. Actually, when $a<b$, local zeta functions can be meromorphically extended to the half-plane ${\rm Re}(s)>-1/a$ and their poles on the half-plane are contained in the set $\{-k/b:k\in\mathbb{N}$ with $k<b/a\}$.