On Some Specific Order Estimates for the Mertens Function and Its Relation to the Non-trivial Zeros of $\zeta(s)$

The primary purpose of this article is to provide a brief overview about the notion of the \textit{Mertens Function} $M(n)$ and \textit{Redheffer Matrices} $\mathbb{A}_{n}$ and, to derive specific order estimates for $M(n)$, in order to analyze the location of the non-trivial zeros of the \textit{Riemann Zeta function} $\zeta(s)$. A priori using the above concepts, the article aims at establishing a necessary and sufficient condition for the \textit{Riemann Hypothesis} to hold true.