In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $ζ(f,χ)$ 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.

The primary purpose of this article is to provide an explicit interpretation of \textit{Ramanujan}'s Partition Congruences, namely, $p(5n+4)\equiv 0\mbox{ }(mod\mbox{ }5)$, $p(7n+5)\equiv 0\mbox{ }(mod\mbox{ }7)$ and, $p(11n+6)\equiv 0\mbox{ }(mod\mbox{ }11)$, $p(n)$ being the \textit{partition function} corresponding to any positive integer $n$, in terms of their corresponding \textit{Cranks} using various combinatorial arguments implemented by \textit{Dyson}, \textit{Atkin}, \textit{Swinnerton-Dyer} and later by \textit{Andrews} and, \textit{Garvan} to justify all the necessary concepts pertaining to this topic. 2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. Secondary 11B75, 05A16, 05A17, 05A30.

This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.