Version 1
: Received: 9 September 2023 / Approved: 12 September 2023 / Online: 12 September 2023 (08:55:16 CEST)
How to cite:
De, S. On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function. Preprints2023, 2023090723. https://doi.org/10.20944/preprints202309.0723.v1
De, S. On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function. Preprints 2023, 2023090723. https://doi.org/10.20944/preprints202309.0723.v1
De, S. On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function. Preprints2023, 2023090723. https://doi.org/10.20944/preprints202309.0723.v1
APA Style
De, S. (2023). On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function. Preprints. https://doi.org/10.20944/preprints202309.0723.v1
Chicago/Turabian Style
De, S. 2023 "On the Order Estimate of the Mertens Function and its Relation to the Zeros of the Riemann Zeta Function" Preprints. https://doi.org/10.20944/preprints202309.0723.v1
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.
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
