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Studying some Inequalities involving Polynomial Functions in π(x)
Version 1
: Received: 15 July 2024 / Approved: 16 July 2024 / Online: 16 July 2024 (08:04:33 CEST)
How to cite: De, S. Studying some Inequalities involving Polynomial Functions in π(x). Preprints 2024, 2024071276. https://doi.org/10.20944/preprints202407.1276.v1 De, S. Studying some Inequalities involving Polynomial Functions in π(x). Preprints 2024, 2024071276. https://doi.org/10.20944/preprints202407.1276.v1
Abstract
This article serves as a extensive research into proposing a few significant inequalities involving polynomial functions in $\pi(x)$, the \textit{prime counting function}, with an intention of exploring the behaviour of $\pi(x)$ for increasing $x$. The general case for such polynomials has also been discussed in detail towards the later section of the article, where the primary focus was to study the order of polynomials of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ being arbitrary polynomials, and establish for a particular case that, the polynomial yields negative values for sufficiently large values of $x$. Furthermore, the error term in such estimations is of order $O\left(\frac{x^d}{(\log x)^{d+1}}\right)$, $d$ depends heavily upon $deg(P)$ and $deg(Q)$.
Keywords
Arithmetic Function; Second Chebyshev Function; Prime Counting Function; Prime Number Theorem; Error Estimates; Polynomials
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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