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- W4386629441 abstract "The primary purpose of this article is to deduce specific order estimates for the textit{Mertens Function} $M(n)$, which facilitates us to analyze the location of zeros of the textit{Riemann Zeta function} $zeta(s)$. The paper also provides a brief overview about the notion of the textit{Mertens Function} $M(n)$ and textit{Redheffer Matrices} $mathbb{A}_{n}$. In addition to learning about various textit{spectral properties} of $ mathbb{A}_{n}$, we shall also deduce the relation between these two, which, eventually would lead us to establish a necessary and sufficient condition for the textit{Riemann Hypothesis} to hold true, as justified by textit{Redheffer} himself. We shall also observe several numerical evidence as well as theoretical justification behind the falsity of the famous textit{Mertens Hypothesis}, along with how researchers over the years have approached towards deriving an estimate of the smallest possible natural number $n$ for which the first such violation of the theorem occurs, utilizing numerous conjectures annotating about the order of $M(n)$. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the texttt{References} section to be extremely resourceful." @default.
- W4386629441 created "2023-09-13" @default.
- W4386629441 creator A5051400442 @default.
- W4386629441 date "2023-09-12" @default.
- W4386629441 modified "2023-09-27" @default.
- W4386629441 title "On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function" @default.
- W4386629441 doi "https://doi.org/10.20944/preprints202309.0723.v1" @default.
- W4386629441 hasPublicationYear "2023" @default.
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