Article
Version 2
This version is not peer-reviewed
Inequalities involving Higher Degree Polynomial Functions in π(x)
Version 1
: Received: 15 July 2024 / Approved: 16 July 2024 / Online: 16 July 2024 (08:04:33 CEST)
Version 2 : Received: 18 July 2024 / Approved: 19 July 2024 / Online: 22 July 2024 (05:19:42 CEST)
Version 2 : Received: 18 July 2024 / Approved: 19 July 2024 / Online: 22 July 2024 (05:19:42 CEST)
How to cite: De, S. Inequalities involving Higher Degree Polynomial Functions in π(x). Preprints 2024, 2024071276. https://doi.org/10.20944/preprints202407.1276.v2 De, S. Inequalities involving Higher Degree Polynomial Functions in π(x). Preprints 2024, 2024071276. https://doi.org/10.20944/preprints202407.1276.v2
Abstract
The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\pi(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $\pi(x)$ and the \textit{Second Chebyshev Function} $\psi(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $\zeta(s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.
Keywords
Arithmetic Function; Second Chebyshev Function; Prime Counting Function; Prime Number Theorem; Error Estimates; Higher-Degree Polynomials; Weighted Sums; Logarithmic Weighted Sums
Subject
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.
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