# Local Zeta Functions, pseudodifferential operators, and Sobolev-type spaces over non-Archimedean Local Fields

## TL;DR

This paper introduces new local zeta functions over non-Archimedean fields, explores their meromorphic properties, and investigates their connection with pseudodifferential operators and Sobolev spaces.

## Contribution

It defines novel local zeta functions involving complex powers of polynomial norms and studies their analytic continuation and relation to pseudodifferential equations.

## Key findings

- Zeta functions admit meromorphic continuation in characteristic zero.
- Poles can have irrational real parts, unlike classical cases.
- Connections established between zeta functions and fundamental solutions of pseudodifferential equations.

## Abstract

In this article we introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.