Meromorphic Continuation Of Global Zeta Function For Number Fields

Abstract

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $\zeta (f,\chi )$ of a Global Number Field $K$ and also deduce the functional equation for the same, using different properties of the idèle class group ${\mathcal{C}}_K^1$ of a global field $K$ extensively defined using basic notions of Adèles (${\mathbb{A}}_K$) and Idèles (${\mathbb{I}}_K$) of $K$, and also evaluating Fourier Transforms of functions $f$ on the space $\mathcal{S}({\mathbb{A}}_K)$ of Adèlic Schwartz-Bruhat Functions. A brief overview of most of the concepts required to prove our desired result have been provided to the readers in the earlier sections of the text.

23 Pages