Abstract

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function ζ(f, X) of a Global Number Field K and also deduce the functional equation for the same, using different properties of the idéle class group C_k¹ of a global field K extensively defined using basic notions of Adéles (A_K) and Idéles (I_K) of K, and also evaluating Fourier Transforms of functions f on the space S(A_K) of Adélic 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.

Keywords: Adéles, Idéles, Global Zeta Function, Archimedean Valuations, NonArchimedean Valuations, Restricted Direct Products, Fourier Transforms, Riemann-Roch Theorem, Characters of a Group, Haar Measure, Schwartz-Bruhat Spaces, Adélic Schwartz-Bruhat Functions, Local Rings, Poisson Summation Formula, Meromorphic Continuation