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 id\`ele class group $\mathcal{C}_K^1$ of a global field $K$ extensively defined using basic notions of Ad\`eles ($\mathbb{A}_{K}$) and Id\`eles ($\mathbb{I}_{K}$) of $K$, and also evaluating Fourier Transforms of functions $f$ on the space $\mathcal{S}(\mathbb{A}_{K})$ of Ad\`elic 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.